Corner asymptotics of the magnetic potential in the eddy-current … · 2017. 1. 1. ·...

Corner asymptotics of the magnetic potential in the

eddy-current model

Monique Dauge, Patrick Dular, Laurent Krahenbuhl, Victor Peron, Ronan

Perrussel, Clair Poignard

To cite this version:

Monique Dauge, Patrick Dular, Laurent Krahenbuhl, Victor Peron, Ronan Perrussel, et al..Corner asymptotics of the magnetic potential in the eddy-current model. Mathematical Meth-ods in the Applied Sciences, Wiley, 2014, 37 (13), pp.1924-1955. <10.1002/mma.2947>. <hal-00779067>

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RESEARCHREPORTN° 8204January 2013

Project-Teams Magique-3D andMC2

Corner asymptotics of the magnetic

potential in the eddy-current model

M. Dauge1, P. Dular2, L. Krähenbühl3, V. Péron4, R. Perrussel5 and C.Poignard61 IRMAR CNRS UMR6625, Rennes, France

2 F.R.S.-FNRS, ACE research unit, Liège, Belgium

3 Laboratoire Ampère CNRS UMR5005, Lyon, France

4 LMAP CNRS UMR5142 & Team M3D INRIA, Université de Pau, Pau, France

5 LAPLACE CNRS UMR5213, Toulouse, France

6 Team MC2, INRIA Bordeaux-Sud-Ouest & CNRS UMR 5251, Bordeaux, France

RESEARCH CENTREBORDEAUX – SUD-OUEST

351, Cours de la Libération

Bâtiment A 29

33405 Talence Cedex

Corner asymptotics of themagnetic potential in the

eddy-current model

M. Dauge1, P. Dular2, L. Krähenbühl3, V. Péron4, R. Perrussel5

and C. Poignard6

1 IRMAR CNRS UMR6625, Rennes, France

2 F.R.S.-FNRS, ACE research unit, Liège, Belgium

3 Laboratoire Ampère CNRS UMR5005, Lyon, France

4 LMAP CNRS UMR5142 & Team M3D INRIA, Université de Pau, Pau, France

5 LAPLACE CNRS UMR5213, Toulouse, France

6 Team MC2, INRIA Bordeaux-Sud-Ouest & CNRS UMR 5251, Bordeaux, France

Project-Teams Magique-3D and MC2

Research Report n° 8204 — January 2013 — 45 pages

Abstract: In this paper, we describe the scalar magnetic potential in the vicinity of a corner ofa conducting body embedded in a dielectric medium in a bidimensional setting. We make explicitthe corner asymptotic expansion for this potential as the distance to the corner goes to zero. Thisexpansion involves singular functions and singular coefficients. We introduce a method for the calcu-lation of the singular functions near the corner and we provide two methods to compute the singularcoefficients: the method of moments and the method of quasi-dual singular functions. Estimates forthe convergence of both approximate methods are proven. We eventually illustrate the theoreticalresults with finite element computations. The specific non-standard feature of this problem lies inthe structure of its singular functions: They have the form of series whose first terms are harmonicpolynomials and further terms are genuine non-smooth functions generated by the piecewise constantzeroth order term of the operator.

Key-words: Corner asymptotics, Calculus of singular functions, Magnetic potential, Eddy currentproblem

Asympotiques de coin pour le potentiel magnétique dans leproblème des courants de Foucault

Résumé : Dans cet article, nous décrivons le potentiel scalaire magnétique solution du problèmebidimensionnel des courants de Foucault au voisinage du coin d’un domaine conducteur immergé dansun milieu diélectrique. Nous explicitons l’asymptotique de coins de ce potentiel lorsque la distance aucoin tend vers zéro. Cette asymptotique nécessite le calcul des fonctions singulières et des coefficientsde singularités. Nous proposons une méthode pour calculer récursivement les fonctions singulières.La méthode quasi-duale est présentée pour le calcul approché des coefficients de singularités et desestimations d’erreur sont prouvées. Nous illustrons ces résultats théoriques par des simulations par élé-ments finis. La caractéristique principale de ce travail réside dans la structure des fonctions singulières:l’inhomogénéité de l’opérateur d’ordre 2 et le fait que son terme d’ordre 0 soit constant par morceauxentraînent que les fonctions singulières sont obtenus sous forme de séries dont chaque premier termeest un polynôme harmonique et dont les termes suivant sont des fonctions analytiques par morceauxgénérés par le terme d’ordre 0.

Mots-clés : Asymptotiques de coins, Calcul de fonctions singulières, Potentiel magnétique, Problèmedes courants de Foucault

4 M. Dauge, P. Dular, L. Krähenbühl, V. Péron, R. Perrussel and C. Poignard

Contents

1 Introduction 51.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Corner asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Laplace operator 82.1 The method of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 The method of dual functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 The problem with angular conductor 103.1 Principles of the construction of the primal singularities . . . . . . . . . . . . . . . . . 123.2 Principles of the construction of the dual singularities . . . . . . . . . . . . . . . . . . 133.3 A first try to extract the singular coefficients: the method of moments . . . . . . . . 143.4 Extraction of singular coefficients by the method of quasi-dual functions . . . . . . . . 16

4 Leading singularities and their first shadows in complex variables 204.1 Formalism in complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Calculation of the shadow function generated by zk and log z . . . . . . . . . . . . . 224.3 Explicit expressions of the first shadows of primal singularities. . . . . . . . . . . . . . 254.4 Explicit expressions of the first shadows of dual singularities . . . . . . . . . . . . . . 26

5 Numerical simulations 27

6 Conclusion 32

References 34

A Appendix: Tools for the derivation of the shadows at any order 35A.1 Definition of the space Jλn on G and the congruence relation ≡ on Sλ`,n and Jλn . . . . . 36A.2 Generic elementary calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A.2.1 The case λ 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.2.2 The case λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A.3 Formal derivation of the singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.3.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.3.2 Generic expression of the shadows at any order . . . . . . . . . . . . . . . . . 44

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Corner asymptotics of the magnetic potential in the eddy-current model 5

1 Introduction

Accurate knowledge of eddy currents is of great interest for the design of many electromagneticdevices such as used in electrothermics. Taking advantage of the fast decrease of the electromagneticfield inside the conductor, impedance conditions are usually considered to reduce the computationaldomain. Impedance conditions were first proposed by Leontovich [13] and by Rytov [17] in the 1940’sand then extended by Senior and Volakis [18]. These conditions can be derived up to the requiredprecision for any conductor with a smooth surface: we refer to Haddar et al. [9] for the mathematicaljustification of conditions of order 1, 2 and 3. Note however that such conditions are not valid nearcorners. Few authors have proposed heuristic impedance modifications close to the corners [6, 21], butthese modifications are neither satisfactory nor proved. In particular in [21], the modified impedanceappears to blow up near the corner, which does not seem valid for non-magnetic materials with a finiteconductivity as presented in [1].

In the present paper, we do not consider the derivation of the impedance condition, which is anasymptotic expansion in high frequency (or high conductivity). We are rather interested in the descrip-tion of the magnetic potential in the vicinity of the corner of a conducting body embedded in a dielectricmedium in a bidimensional setting, considering the conductivity σ, as well as the angular frequencyκ, as given parameters. We emphasize that these considerations will be helpful in understanding thebehavior of the impedance condition near corner singularities that will be considered in a forthcomingpaper.

Roughly speaking, the aim of the present paper is to provide the asymptotic expansion of themagnetic potential as the distance to the corner goes to zero. This notion of corner asymptoticsgeneralizes the Taylor expansion, which holds in smooth domains. Such asymptotics involve two mainingredients:

• The singular functions, also called singularities, which belong to the kernel of the consideredoperator.

• The singular coefficients, whose calculation requires the knowledge of (quasi-) dual singularfunctions.

In the present paper, we consider the magnetic potential in a non-magnetic domain composed of aconducting material Ω− with one corner surrounded by a dielectric material Ω+. Thus the operatoracting on the magnetic potential is not homogeneous and has a discontinuous piecewise constantcoefficient in front of its zeroth order part:

−∆ + iκµ0σ1Ω− .

Though pertaining to the wide class of elliptic boundary value problems or transmission problems inconical domains, see for instance the papers [10, 11] and the monographs [8, 5, 16, 12], this problemhas specific features which make the derivation of the asymptotics not obvious—namely the fact thatsingularities are generated by a non-principal term. Despite its great interest for the applications, thisproblem has not yet been explicitly analyzed.

Another disturbing factor is the nature of the limit problem when the product κµ0σ tends to infinity(large frequency/high conductivity limit). This limit is simply the homogeneous Dirichlet problem forthe Laplace operator set in the dielectric medium Ω+, whose corner singularities are well known [10, 8].In particular, when the conductor Ω− has a convex corner, its surrounding domain Ω+ has a non-convexcorner, thus the Dirichlet problem has non C1 singularities, in opposition to the problem for any finiteκµ0σ. This apparent paradox can be solved by a delicate multi-scale analysis, whose heuristics are

RR n° 8204


exposed in [1]. Roughly speaking, there exists profiles in R2 which have the singular behavior of theoperator −∆ + 2i1Ω− near the corner, and which connect at infinity with the singular functions ofthe Dirichlet problem in Ω+, as described by equation (13) in [1]. This is why the knowledge of thesingularities of −∆ + iκµ0σ1Ω− at given κ and σ is a milestone in the full multi-scale analysis.

Besides the mere description of singular functions, the computation of singular coefficients is con-sidered in this paper. Various works concern the extraction of singular coefficients associated with thesolution to an elliptic problem set in a domain with a corner singularity, see [14, 4] for theoretical for-mulas, and [15, 20, 3] for more practical methods. Here, we choose to extend the quasi-dual functionmethod initiated in [3] to the case of resonances, which was discarded in the latter reference.

Let us now present the problem considered throughout the paper.

1.1 Statement of the problem

Denote by Ω− ⊂ R2 the bounded domain corresponding to the conducting medium, and by Ω+ ⊂ R2

the surrounding dielectric medium (see Figure 1(a)). The domain Ω with a smooth boundary Γ isdefined by Ω = Ω− ∪ Ω+ ∪ Σ, where Σ is the boundary of Ω−. For the sake of simplicity, we assumethat:

1. Σ has only one geometric singularity, and we denote by c this corner. The angle of the corner(from the conducting material, see Figure 1(a)) is denoted by ω.

2. The current source term J, located in Ω+, is a smooth function, which vanishes in a neighborhoodof c.

Γ

Ω+

Σ

c Ω−ω

(a) Domain for the eddy-current problem.

r S−S+

G

θ = 0

θ = ω/2

θ = −ω/2

(b) Infinite sectors S+ and S− in R2.

Figure 1: Geometries of the considered problems.

The magnetic vector potential A (reduced to a single scalar component in 2d) satisfies the followingproblem:

−∆A+ = µ0J in Ω+,

−∆A− + iκµ0σA− = 0 in Ω−,

A+ = 0 on Γ,

[A]Σ = 0, on Σ,

[∂nA]Σ = 0, on Σ,

where i is the imaginary unit, µ0 the vacuum permeability, κ the angular frequency of the phenomenonand σ the electric conductivity. Here, A± denotes the restriction of A to Ω± and [A]Σ := A+|Σ−A−|Σ

Inria


is the jump of A across the interface Σ. Similarly, [∂nA]Σ is the jump of the normal derivative of A,where n is the unit normal vector to Σ. Since we consider the parameters as given in the presentpurpose, for the sake of normalization we introduce the positive parameter ζ as

ζ2 = κµ0σ/4,

so that the problem satisfied by A writes−∆A+ = µ0J in Ω+,

−∆A− + 4iζ2A− = 0 in Ω−,

A+ = 0 on Γ,

[A]Σ = 0, on Σ,

[∂nA]Σ = 0, on Σ.(1)

The factor 4 is present in order to simplify the forthcoming calculations (see equation (47)) in Section4 and in Appendix-A.

The variational formulation of problem (1) isFind A ∈ H1

0(Ω) such that ∀A∗ ∈ H10(Ω),∫

Ω+

∇A+ ·∇A+∗ dxdy +

∫Ω−

∇A− ·∇A−∗ + 4iζ2A−A−∗ dxdy = µ0

∫Ω

JA∗ dxdy .

Since this form is coercive on H10(Ω), there exists a unique solution A in H1

0(Ω) to problem (1). Inaddition, note that A is solution to the Dirichlet problem

−∆A = F in Ω,

A = 0 on Γ,(2)

where F = µ0J 1Ω+ −4iζ2A1Ω− , the functions 1Ω± being the characteristic functions of Ω±. Since Jis smooth and since A belongs to H1(Ω), then F belongs to L2(Ω), and even to H

12−ε(Ω) for any ε > 0,

— here the bound on the Sobolev exponent comes from the discontinuity of F across Σ. Standardelliptic regularity of the Laplace operator implies that A belongs to H2(Ω), and even to H

52−ε(Ω) for

any ε > 0. In particular, A belongs to C1(Ω) thanks to Sobolev imbeddings in two dimensions.

1.2 Corner asymptotics

In general, the magnetic potential A does not belong to C2(Ω), but the function A possesses a cornerasymptotic expansion as the distance r to c goes to zero.

In contrast with problems involving only homogeneous operators, problem (1) involves the lowerorder term 4iζ2 in Ω−. As a consequence, according to the seminal paper of Kondratev [10], thesingularities are not homogeneous functions, but infinite sums of quasi-homogeneous terms of thegeneral form

rλ+` lognr Φ(θ), λ ∈ C, ` ∈ N, n ∈ N

in polar coordinates (r, θ) centered at c. In the present situation, the leading exponents λ can bemade precise: they are determined by the principal part of the operator at c, which is nothing but theLaplacian −∆ at the interior point c. Thus, the leading exponents are integers k ∈ N correspondingto leading singularities in the form of harmonic polynomials, written in polar coordinates as

r k cos(kθ−pπ/2), k ∈ N, p ∈ 0, 1, (3)

RR n° 8204


— only p = 0 is involved when k = 0. This is why A can be expanded close to the corner as

A(r, θ) ∼r→0

Λ0,0S0,0(r, θ) +∑k>1

∑p∈0,1

Λk,pSk,p(r, θ), (4)

where the terms Sk,p are the so-called primal singular functions, which belong to the formal kernel ofthe considered operator in R2, and the numbers Λk,p are the singular coefficients.

Therefore, the derivation of the singular functions Sk,p and the determination of the singularcoefficients Λk,p are key points in understanding the behavior of the magnetic potential in the vicinityof the corner. This is the double aim of this work.

1.3 Outline of the paper

In this paper, we provide a constructive procedure to determine the singular functions Sk,p as wellas two different methods to compute the singular coefficients Λk,p. Generally speaking, the singularfunctions are sums of the kernel functions of the Laplacian plus shadow terms. Since the leading part ofthe operator is the Laplacian in the whole plane R2, its singular functions are the harmonic polynomials(3) and we show that for any (k, p) ∈ N× 0, 1, the singular function Sk,p writes as formal series

Sk,p(r, θ) = r k∑j>0

(iζ2)j r2j

j∑n=0

lognr Φk,pj,n (θ),

where the angular functions Φk,pj,n are C1 functions of θ globally in R/(2πZ) (and piecewise analytic).

The paper is organized as follows: In Section 2, we consider the case when ζ = 0 and show howthe theory of corner expansions applies to the Taylor expansion of A in the interior point c. For thedetermination of the coefficients Λk,p, we introduce the method of moments and the method of dualsingular functions, which are both exact methods in this case. In Section 3, for the case ζ 6= 0, weprovide the problem satisfied by the singular functions Sk,p in the whole plane R2 split into two infinitesectors S± (cf. Figure 1(b)). In order to determine the coefficients Λk,p, we generalize the method ofmoments, and we introduce the method of quasi-dual singular functions, after [3, 19]. These methodsare now approximate methods. We prove estimates for their convergence. In section 4, we introducea formalism using complex variables z± along the lines of [2], which simplifies the analytic calculationsof the primal singular functions Sk,p and the dual singular functions Kk,p, required by the quasi-dualfunction method. These dual singularities are composed of the kernel functions of the Laplacian withnegative integer exponents, plus their shadows, quite similarly to the Sk,p. In this fourth section, onlythe first order shadows of the kernel functions of the Laplacian are given, while Appendix A providestools of calculus of the shadows at any order. In the concluding Section 5, numerical simulations byfinite elements methods illustrate the theoretical results.

2 Laplace operator

We first consider the case when ζ = 0, which means that we consider the solution A to the Laplaceequation (2) with a smooth right-hand side F , whose support is located outside the interior point c,and we look at the Taylor expansion of A at c. Since A is harmonic in a neighborhood of c, all theterms of its Taylor expansion are harmonic polynomials. A basis of harmonic polynomials can be writtenin polar coordinates as

sk,p(r, θ) =

1, if k = 0 and p = 0,

r k cos(kθ−pπ/2), if k > 1 and p = 0, 1,(5)

Inria


and A admits the Taylor expansion

A(r, θ) ∼r→0

Λ0,0s0,0(r, θ) +∑k>1

∑p∈0,1

Λk,psk,p(r, θ). (6)

There are several methods to extract the coefficients Λk,p. Besides taking point-wise values of partialderivatives (which would be unstable from the numerical approximation viewpoint), we can considertwo methods:

1. The method of moments, which uses the orthogonality of the angular parts cos(kθ−pπ/2) ofthe polynomials sk,p.

2. The dual function method, which is the application to the present smooth case of a generalmethod valid for corner coefficient extraction [14].

Both methods use non-polynomial dual harmonic functions singular at r = 0. Let us set

kk,p(r, θ) =

−

1

2πlog r, if k = 0, p = 0,

1

2kπr−k cos(kθ−pπ/2), if k > 1, p = 0, 1.

(7)

2.1 The method of moments

To extract the singular coefficients with the method of moments, we introduce the symmetric bilinearformMR defined for R > 0 by

MR(K,A) :=1

R

∫r=R

K A R dθ. (8)

We extract the scalar product of A versus 1 to compute Λ0,0 and versus kk,p to compute Λk,p whenk > 1.

Proposition 2.1. Let A be the solution to the Laplace equation (2) with a smooth right-hand side Fwith support outside the ball B(c, R). Then

MR(1,A) = 2πΛ0,0, and MR(kk,p,A) =1

2kΛk,p, for k > 1, p = 0, 1. (9)

Proof. The first equality is straightforward since

MR(1, sl ,p) = 0, l > 1, p = 0, 1.

Using the orthogonality between the polynomials sl ,q and the dual functions kk,p we infer

MR(kk,p, sl ,q) =1

2kδkl δ

pq , k > 1, l > 0, p, q = 0, 1, (10)

where δji denotes the Kronecker symbol. Then, Taylor expansion (6) of A leads to the second equality(9).

RR n° 8204


2.2 The method of dual functions

Let us introduce the anti-symmetric bilinear form JR defined over a circle of radius R > 0 by

JR(K,A) :=

∫r=R

(K∂rA− A∂rK) Rdθ. (11)

Proposition 2.2. Let A be the solution to the Laplace equation (2) with a smooth right-hand side F ,whose support is located outside the ball B(c, R). Then

JR(k0,0,A) = Λ0,0, and JR(kk,p,A) = Λk,p, for k > 1, p = 0, 1. (12)

Proof. Since JR(k0,0, sk,p) = 0 for k 6= 0, there holds

JR(k0,0,A) = JR(k0,0,Λ0,0) = Λ0,0 .

Straightforward computations lead to

JR(kk,p, sl ,q) = lMR(kk,p, sl ,q) + kMR(kk,p, sl ,q), k, l > 1, p, q = 0, 1.

Then, using (10) and (6), we obtain

JR(kk,p,A) = Λk,p, for k > 1, p = 0, 1 ,

which ends the proof.

3 The problem with angular conductor

We now consider our problem of interest, i.e. problem (1) with ζ 6= 0. Let ω ∈ (0, 2π) be the openingof the conductor part Ω− near its corner. We define the origin of the angular variable, so that θ = 0

cuts the conductor part by half (see Figure 1(b)). Let S+ and S− be the two infinite sectors of R2

defined as

S− =

(x, y) = (r cos θ, r sin θ) ∈ R2 : r > 0, θ ∈ (−ω/2, ω/2), S+ := R2 \ S− .

We denote by G the common boundary of S− and S+:

G = ∂S− = ∂S+ =

(x, y) = (r cos θ, r sin θ) ∈ R2 : r > 0, θ = −ω

2,ω

2

.

Finally, introduce the unit circle T = R/(2πZ) and set

T− = (−ω/2, ω/2) and T+ = T \ T− .

Thus, the configuration (Ω−,Ω+) coincides with the configuration (S−,S+) inside a ball B(c, R) for asufficiently small R.

The positive parameter ζ being chosen, we denote by Lζ the operator defined on R2 by

Lζ(u) =

−∆u, in S+,

−∆u + 4iζ2u, in S−.(13)

acting on functions u such that ∂qθ u−|G = ∂qθ u+|G for q = 0, 1. The singularities of problem (1) arethese of the model operator Lζ, with continuous Dirichlet and Neumann traces across G.

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As described above, the asymptotics of A near the corner involve the singularities of Lζ which, byconvention, are formal solutions U to the equation Lζ(U) = 0. It is therefore crucial to make explicitthese singularities.

Note that the operator Lζ is the sum of its leading part L0 which is the Laplacian −∆ in R2, andof its secondary part 4iζ2L1 where L1 is the restriction to S−. According to the general principles ofKondratev’s paper [10], the singularities U of Lζ = L0 + 4iζ2L1 can be described as formal sums

U =∑j

(iζ2)juj , (14)

where each term uj is derived through an inductive process by solving recursively

L0u0 = 0, L0u1 = −4L1u0, . . . , L0uj = −4L1uj−1, (15)

in spaces of quasi-homogeneous functions Sλ defined for λ ∈ C by

Sλ = Spanrλ logqr Φ(θ), q ∈ N, Φ ∈ C1(T), Φ± ∈ C∞(T±)

. (16)

The numbers λ are essentially determined by the leading equation L0u0 = 0, whose solutions areharmonic functions in R2 \ c. The first term u0 is the leading part of the singularity while the nextterms uj , for j > 1, are called the shadows.

The existence of the terms uj relies on the following result.

Lemma 3.1. For λ ∈ C, let Sλ be defined by (16) and let Tλ be defined as

Tλ = Spanrλ logqr Ψ(θ), q ∈ N, Ψ ∈ L2(T), Ψ± ∈ C∞(T±)

. (17)

For an element g of Sλ or Tλ, the degree deg g of g is its degree as polynomial of log r .Let λ ∈ C and f ∈ Tλ−2. Then, there exists u ∈ Sλ such that L0u = f. Moreover

(i) If λ 6∈ Z, deg u = deg f,

(ii) If λ ∈ Z \ 0, deg u 6 deg f + 1,

(iii) If λ = 0, deg u 6 deg f + 2.

Proof. We follow the technique of [5, Ch. 4]. We introduce the holomorphic family of operators M(also called Mellin symbol) associated with L0 in polar coordinates

L0 = −∆ = −r−2((r∂r )2 + ∂2

θ ) and M (µ) = −(µ2 + ∂2θ ), θ ∈ T.

Note the following facts

i) The poles of M (µ)−1 are the integers,

ii) If k is a nonzero integer, the pole of M (µ)−1 in k is of order 1,

iii) The pole of M (µ)−1 in 0 is of order 2.

Let f ∈ Tλ−2. Then

f =

deg f∑q=0

rλ−2 logqr Ψq(θ).

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We have the residue formula1

f = Resµ=λ

rµ−2

deg f∑q=0

q! Ψq

(µ− λ)q+1

.

Setting

u = Resµ=λ

rµM (µ)−1

deg f∑q=0

q! Ψq

(µ− λ)q+1

,

we check that L0u = f and the assertions on deg u are consequences of the above properties i), ii) andiii).

At this point, we distinguish the primal singular functions, which belong to H1 in any bounded neigh-borhood B of c, from the dual singular functions, which do not belong to H1 in such a neighborhood.The primal singular functions S appear in the expansion as r → 0 of the solutions to problem (1),while the dual singular functions K are needed for the determination of the coefficients Λ involved inthe asymptotics.

3.1 Principles of the construction of the primal singularities

The primal singular functions start with (quasi-homogeneous) terms u0 ∈ H1(B) satisfying ∆u0 = 0.Therefore, u0 is a homogeneous harmonic polynomial. We are looking for a basis for primal singularfunctions, so we choose u0 = sk,p (for (k, p) = (0, 0) and for any (k, p) ∈ N∗ × 0, 1) according tonotation (5). In order to have coherent notation, we set sk,p0 := sk,p. Each sk,p0 belongs to Sk and isthe leading part of the singular function Sk,p defined by

Sk,p(r, θ) =∑j>0

(iζ2)jsk,pj (r, θ). (18)

For j > 1, the terms sk,pj in the previous series are the shadow terms of order j . According to the

sequence of problems (15), the function sk,pj is searched in Sk+2j as a particular solution to the followingproblem:

∆sk,p+j = 0, in S+ ,

∆sk,p−j = 4sk,p−j−1 , in S− .(19)

Belonging to Sk+2j includes the following transmission conditions on G:sk,p+j |G = sk,p−j |G ,

∂θsk,p+j |G = ∂θs

k,p−j |G .

(20)

Lemma 3.2. Let (k, p) = (0, 0) or (k, p) ∈ N∗ × 0, 1. Let sk,p0 be defined as

sk,p0 (r, θ) =

1, if k = 0 and p = 0,

r k cos(kθ−pπ/2), if k > 1 and p = 0, 1 .

Then, for any j > 1, there exists sk,pj ∈ Sk+2j satisfying (19)-(20). Moreover deg sk,pj 6 j . When

p = 0, we choose sk,pj as an even function2 of θ and when p = 1, sk,pj is chosen to be odd.

1For a meromorphic function f of the complex variable µ with pole at λ and for any simple contour C surrounding λ

and no other pole of f , there holds Resµ=λf (µ) =

1

2iπ

∫Cf (µ) dµ .

2We understand that u is an even function of θ if u(−θ) = u(θ) for all θ ∈ R/(2πZ).

Inria


Proof. We apply recursively Lemma 3.1. We start with sk,p0 , which belongs to Sk and satisfies deg sk,p0 =

0. The right-hand side of (19) for j = 1 is 4sk,p0 1S− . It is discontinuous across G, belongs to Tk ,and is even (resp. odd) for p = 0 (resp. p = 1). By Lemma 3.1 there exists sk+2,p

1 ∈ Sk+2 solutionto (19)-(20) for j = 1, and deg sk+2

1 6 1. Since ∆ commutes with the symmetry with respect to xaxis, there exists a particular solution with the same parity as the right-hand side. The proof for any jfollows from a recursion argument.

These definitions being set, we can write the expansion of the solution to problem (1) as

A(r, θ) ∼r→0

Λ0,0S0,0(r, θ) +∑k>1

∑p∈0,1

Λk,pSk,p(r, θ). (21)

Moreover, using the results of Lemma 3.2 and (16) in (18), it is straightforward that for any (k, p) ∈N∗ × 0, 1 or (k, p) = (0, 0), the singular function Sk,p writes as formal series

Sk,p(r, θ) = r k∑j>0

(iζ2)j r2j

j∑n=0

lognr Φk,pj,n (θ), (22)

where the angular functions Φk,pj,n are C1 functions of θ globally in R/(2πZ) (and piecewise analytic).

3.2 Principles of the construction of the dual singularities

We start with the dual singularities of the interior Laplace operator introduced in (7) . Setting kk,p0 =

kk,p, the dual singularities of Lζ are given by series

Kk,p(r, θ) =∑j>0

(iζ2)jkk,pj (r, θ), (23)

where, for j > 1, the shadow terms kk,pj are solutions in S−k+2j to∆kk,p+

j = 0, in S+ ,

∆kk,p−j = 4kk,p−j−1 , in S− ,(24)

with the same transmission conditions as (20). We deduce from Lemma 3.1:

Lemma 3.3. Let (k, p) = (0, 0) or (k, p) ∈ N∗ × 0, 1. Let kk,p0 be defined as

kk,p0 (r, θ) =

−

1

2πlog r, if k = 0, p = 0,

1

2kπr−k cos(kθ−pπ/2), if k > 1, p = 0, 1 .

Then, for any j > 1, there exists kk,pj ∈ S−k+2j satisfying (24). Moreover, if j is odd, or if j is even and

2j < k , then deg kk,pj 6 j . Otherwise (i.e. if j is even and 2j > k) then deg kk,pj 6 j + 1. When p = 0,

we choose kk,pj as an even function of θ and when p = 1, kk,pj is chosen to be odd.

RR n° 8204


Using the results of Lemma 3.3 and (16) in (23), it is straightforward that for any (k, p) ∈N∗ × 0, 1 or (k, p) = (0, 0), the singular function Kk,p writes as formal series

Kk,p(r, θ) =

r−k

∑j>0

(iζ2)j r2j

j∑n=0

lognr Ψk,pj,n (θ) (k odd),

r−k( ∑

06j<k/2

(iζ2)j r2j

j∑n=0

lognr Ψk,pj,n (θ) +

∑j>k/2

(iζ2)j r2j

j+1∑n=0

lognr Ψk,pj,n (θ)

)(k even).

(25)where the angular functions Ψk,p

j,n are C1 functions of θ globally in R/(2πZ) (and piecewise analytic).

3.3 A first try to extract the singular coefficients: the method of moments

A naive approach to extract the coefficients consists in following the heuristics of the case when ζ = 0,cf. (8)-(9): compute MR(1/(2π),A) in order to extract the coefficient Λ0,0 and MR(2kkk,p,A) forthe coefficient Λk,p of expansion (21). This method makes possible a quite rough approximation ofthe first coefficients only, as described below. For a systematic computation of all the coefficients, themethod of quasi-dual functions, described in the next section is more accurate.

Let us set

Mk,pR (A) =

1

2πMR(1,A) if k = 0 and p = 0,

2kMR(kk,p0 ,A) if k > 1 and p = 0, 1.

We input the expansion (21) of A in Mk,pR to obtain the formal expression

MR(A) =MR Λ(A),

withMR(A) and Λ(A) the column vectors (M0,0R (A),M1,0

R (A),M1,1R (A), . . .)> and (Λ0,0,Λ1,0,Λ1,1, . . .)>,

respectively, andMR the matrix with coefficients

Mk,p ; k ′,p′

R = Mk,pR (Sk ′,p′).

Hence, we deduce the formal expression of the coefficients Λk,p

Λ(A) =(MR

)−1MR(A).

In order to give a sense to this expression, and to design a method for the determination of thecoefficients Λk,p, we have to truncate the matrix MR by taking its first N rows and columns, N =

1, 2, . . .

Let us denote for shortness

R0 = ζR (1 +√| logR|) (26)

and assume that R 6 1. Note that, relying on Proposition 2.1, Lemma 3.2 and formula (18), we find

Mk,p ; k,pR =

R→01 +O

(R2

0

)and Mk,p ; k ′,p′

R = 0 if p 6= p′.

We also immediately check that

Mk,p ; k ′,p′

R =R→0O(R−k+k ′R2

0

)if k 6= k ′.

Inria


Let us emphasize the blow up of this coefficient as R→ 0 if k > k ′+ 2, which prevents the use of thismethod to approximate the singular coefficients at any level of accuracy. This drawback is avoided bythe method of the quasi-dual functions presented in the next subsection.

However, due to its simplicity, the method of moments can be useful if a low order of accuracy ofthe first 3 coefficients is sufficient. In the following, we list what can be deduced from the considerationof the lowest values for the cut-off dimension N.

• N = 1. We consider the sole approximate equation

M0,0R (A) = M0,0

R (S0,0) Λ0,0 +O(RR2

0

).

The remainder O(RR2

0

)corresponds to the neglected terms in the asymptotics of A. Using Lemma

3.2 for the structure of S0,0, we find

S0,0(r, θ) =r→0

1 + i(ζr)21∑q=0

logqr Φ0,01,q(θ) +O((ζr)4 log2 r). (27)

Therefore we deduce

i) Using only the first term in the asymptotics of S0,0, we find

Λ0,0 =R→0

M0,0R (A) +O

(R2

0

). (28)

ii) If we compute 12π

∫TΦ0,0

1,q(θ) dθ =: m0,0 ; 0,01,q , we find

Λ0,0 =R→0

M0,0R (A)

1 + i(ζR)2∑1

q=0 m0,0 ; 0,01,q logqR

+O(RR2

0

)+O

(R4

0

). (29)

iii) If we compute more terms in the asymptotics of S0,0, we improve the piece of error O(R4

0

), but

leave the piece O(RR2

0

)unchanged.

• N = 3. We consider the 3 approximate equations concerning M0,0R (A), M1,0

R (A) and M1,1R (A). For

parity reasons, we have a 2× 2 system

M0,0R (A) = M0,0

R (S0,0) Λ0,0 +M0,0R (S1,0) Λ1,0 +O

(R2R2

0

),

M1,0R (A) = M1,0

R (S0,0) Λ0,0 +M1,0R (S1,0) Λ1,0 +O

(RR2

0

),

and an independent equation

M1,1R (A) = M1,1

R (S1,1) Λ1,1 +O(RR2

0

).

For this latter equation, we have a similar analysis as for Λ0,0 alone (N = 1): Using again Lemma 3.2for the structure of S1,1, we find

S1,1(r, θ) =r→0

r sin θ + i(ζr)2r

1∑q=0

logqr Φ1,11,q(θ) +O((ζr)4r log2 r), (30)

and with the first term in the asymptotics of S1,1, we obtain

Λ1,1 =R→0

M1,1R (A) +O

(R2

0

), (31)

RR n° 8204


whereas, using two terms in the asymptotics of S1,1, and computing 12π

∫T sin θΦ1,1

1,q(θ) dθ =: m1,1 ; 1,11,q ,

we find

Λ1,1 =R→0

M1,1R (A)

1 + i(ζR)2∑1

q=0 m1,1 ; 1,11,q logqR

+O(RR2

0

)+O

(R4

0

). (32)

For the 2 × 2 system, according to the number of terms considered in the asymptotics (18) of S0,0

and S1,0 we find the following formulas.

i) If we take one term for S0,0 in (27) and one term for S1,0 in

S1,0(r, θ) =r→0

r cos θ +O((ζr)2r log2 r),

we obtain a diagonal system and

Λ0,0 =R→0

M0,0R (A) +O

(R2

0

)and Λ1,0 =

R→0M1,0R (A) +O

(R−1R2

0

).

ii) If we take two terms for S0,0 and one for S1,0, we obtain (28) again, and computing m1,0 ; 0,01,q

as 1π

∫T cos θΦ0,0

1,q(θ) dθ, we find

Λ1,0 =R→0

M1,0R (A)−M0,0

R (A)iR−1(ζR)2

∑1q=0 m

1,0 ; 0,01,q logqR

1 + i(ζR)2∑1

q=0 m0,0 ; 0,01,q logqR

+O(RR2

0

)+O

(R−1R4

0

).

iii) Taking more terms, we improve some parts of the remainder, we still have O(R2R2

0

)for Λ0,0,

and O(RR2

0

)for Λ1,0, but we always have a term containing R−1, preventing the good accuracy

of the method.

3.4 Extraction of singular coefficients by the method of quasi-dual functions

When ζ 6= 0, and in contrast to the case ζ = 0, we do not use exact dual functions satisfying LζK = 0

inside the anti-symmetric bilinear form JR (11). Instead we use the quasi-dual functions Kk,pm , whichare the truncated series of (23):

Kk,pm (r, θ) :=

m∑j=0

(iζ2)jkk,pj (r, θ). (33)

Here, m is a nonnegative integer, which is the order of the quasi-dual function. By construction

LζKk,pm = 4iζ2(iζ2)mkk,pm 1S− , (34)

which is not zero, but smaller and smaller as r → 0 when m is increased. The extraction of coefficientsΛk,p in expansion (21) is performed through the evaluation of quantities

JR(Kk,pm ,A), k = 0, 1, 2, . . .

and corresponding p ∈ 0, 1, for suitable values of m ∈ 0, 1, . . .. The quasi-dual function methodwas introduced in [3] for straight edges and developed in [19] for circular edges and homogeneousoperators with constant coefficients. The expansions considered there do not contain any logarithmicterms. Here, we revisit this theory in our framework where, on the contrary, we have an accumulationof logarithmic terms. The main result follows.

Inria


Theorem 3.4. Let A be the solution to problem (1), under the assumptions of the introduction. Letk ∈ N and p ∈ 0, 1 (p = 0 if k = 0). Let m such that 2m + 2 > k . For the extraction quantityJR(Kk,pm ,A) defined through (11) and (33), there exist coefficients J k,p;k ′,p′ independent of R and Asuch that

JR(Kk,pm ,A) =R→0

Λk,p +

[k/2]∑`=1

J k,p ; k−2`,pΛk−2`,p +O(R−kR2m+20 logR) , (35)

where R0 is defined in (26). If p = 1, the remainder is improved to O(R1−kR2m+20 logR). The extra

term logR disappears if k is odd.

Remark 3.5. The collection of equations (35) for p ∈ 0, 1, and k belonging to 0, 2, . . . , 2L or to1, 3, . . . , 2L+ 1, with L ∈ N, forms a lower triangular system with invertible diagonal.

Example 3.6. i) We have the following simple formulas for m = 0:• For k = 0

Λ0,0 =R→0

JR(K0,00 ,A) +O(R2

0 logR) .

• For k = 1

Λ1,0 =R→0

JR(K1,00 ,A) +O(R−1R2

0) and Λ1,1 =R→0

JR(K1,10 ,A) +O(R2

0).

If we know m more terms in the quasi-dual functions, then O(R20) is replaced by O(R2+2m

0 ) in theremainder terms.ii) For k = 2, we need m > 1. We have

Λ2,1 =R→0

JR(K2,1m ,A) +O(R−1R2+2m

0 logR),

for the odd singularity, and

Λ2,0 =R→0

JR(K2,0m ,A)− J 2,0 ; 0,0Λ0,0 +O(R−2R2+2m

0 logR),

for the even singularity. An explicit formula for the coefficient J 2,0 ; 0,0 is given later on, see (65), as aparticular case of the general formula (44).

Proof of Theorem 3.4. Introduce the truncated series of singularities (18):

Sk,pm (r, θ) :=

m∑j=0

(iζ2)jsk,pj (r, θ).

We haveLζS

k,pm = 4iζ2(iζ2)msk,pm 1S− . (36)

In order to prove formula (35), let us evaluate JR(Kk,pm ,Sk ′,p′

m′ ) with any chosen k ′, p′ and m′ > m.For parity reasons, JR(Kk,pm ,Sk ′,p′

m′ ) is zero if p 6= p′. Therefore, from now on we take p′ = p.For ε ∈ (0, R), denote by C(ε,R) be the annulus (x, y) : r ∈ (ε,R). Thanks to Green formula

the following equality holds:

JR(Kk,pm ,Sk ′,pm′ )− Jε(Kk,pm ,Sk ′,p

m′ ) =

∫C(ε,R)

(Kk,pm LζS

k ′,pm′ −LζK

k,pm Sk ′,p

m′

)rdrdθ,

RR n° 8204


hence, by formulas (34) and (36), we deduce

JR(Kk,pm ,Sk ′,p′

m′ )− Jε(Kk,pm ,Sk ′,pm′ ) = 4iζ2

∫C(ε,R)

(Kk,pm (iζ2)m

′sk′,pm′ − (iζ2)mkk,pm Sk ′,p

m′

)1S− rdrdθ.

Using Lemmas 3.2 and 3.3, and taking the condition −k + 2m + 2 > 0 into account, we find


m′ ) =R→0O(R−k+k ′R2m′+2

0 ) +O(R−k+k ′R2m+20 logR),

independently of ε < R, — the term logR can be omitted if k is odd. Hence, since m′ > m it yields:


m′ ) =R→0O(R−k+k ′R2m+2

0 logR) (37)

=R→0

ζ2m+2O(R−k+k ′+2m+2 logm+2R). (38)

We use Lemmas 3.2 and 3.3 to find the general form of Jρ(Kk,pm ,Sk ′,pm′ ) for any ρ > 0. We obtain

Jρ(Kk,pm ,Sk ′,pm′ ) =

m∑j=0

m′∑j ′=0

j+1∑q=0

j ′∑q′=0

J k,p ; k ′

j,q,j ′,q′ (iζ2)j+j′ρ−k+k ′+2j+2j ′ logq+q′ρ (39)

=

m+m′∑j=0

j+1∑q=0

J k,p ; k ′

j ,q (iζ2)j ρ−k+k ′+2j logq ρ. (40)

Fix R > 0 and let ε go to 0. The expression (38) yields that JR(Kk,pm ,Sk ′,pm′ )−Jε(Kk,pm ,Sk ′,p

m′ ) is boundedindependently of ε < R, which means in particular that Jε(Kk,pm ,Sk ′,p

m′ ) is bounded independently ofε < R. Considering (40) for ρ = ε, we deduce that the coefficients J k,p ; k ′

j ,q attached to unbounded

terms ε−k+k ′+2j logq ε, namely such that −k + k ′ + 2j < 0, or −k + k ′ + 2j = 0 and q > 0, are zero.Hence, setting ν = 1

2 (k − k ′), we find


(iζ2)νJ k,p ; k ′

ν,0 +

m+m′∑j=ν+1

j+1∑q=0

J k,p ; k ′

j ,q (iζ2)j ρ−k+k ′+2j logq ρ if ν ∈ N,

m+m′∑j=[ν]+1

j+1∑q=0

J k,p ; k ′

j ,q (iζ2)j ρ−k+k ′+2j logq ρ if ν 6∈ N .

(41)

We note that limε→0 Jε(Kk,pm ,Sk ′,pm′ ) exists and is (iζ2)νJ k,p ; k ′

ν,0 if ν ∈ N and 0 if not. Letting ε→ 0 in(37)-(38) we find in all cases

m+m′∑j=[ν]+1

j+1∑q=0

J k,p ; k ′

j ,q (iζ2)j R−k+k ′+2j logq R =R→0

ζ2m+2O(R−k+k ′+2m+2 logm+2R) .

Therefore the coefficients J k,p ; k ′

j ,q are zero if −k+k ′+ 2j < −k+k ′+ 2m+ 2, i.e. if j < m+ 1. Finally



ν,0 +

m+m′∑j=m+1

j+1∑q=0

J k,p ; k ′

j ,q (iζ2)j ρ−k+k ′+2j logq ρ if ν ∈ N,

m+m′∑j=m+1

j+1∑q=0

J k,p ; k ′

j ,q (iζ2)j ρ−k+k ′+2j logq ρ if ν 6∈ N.

(42)

Inria


Taking the largest term in the remainders of equality (42), we find

JR(Kk,pm ,Sk ′,pm′ ) =

R→0


ν,0 +O(

(iζ2)m+1R−k+k ′+2m+2 logm+2R)

if ν ∈ N,

O(

(iζ2)m+1R−k+k ′+2m+2 logm+2R)

if ν 6∈ N.(43)

Recall that ν = 12 (k−k ′). The case ν ∈ N amounts to considering k ′ = k−2` with ` 6 [k/2]. Setting

J k,p ; k−2`,p = (iζ2)`J k,p ; k−2``,0 ,

and J k,p ; k ′,p′ = 0 otherwise, we have proved the theorem.

Proposition 3.7. Let k ∈ N \ 0, 1 and p ∈ 0, 1. For any ` ∈ 1, . . . , [k/2], the coefficientsJ k,p ;k−2`,p introduced in (35) are given by

J k,p ; k−2`,p = (iζ2)`∑j=0

∫ 2π

0

Ψk,pj,0 (θ)

[2(k − 2j)Φk−2`,p

`−j,0 (θ) + Φk−2`,p`−j,1 (θ)

]−Ψk,p

j,1 (θ)Φk−2`,p`−j,0 (θ) dθ,

(44)with the convention that Φk−2`,p

0,1 = 0 and Ψk,p0,1 = 0.

Proof. Recall that J k,p ; k−2`,p = (iζ2)`J k,p ; k−2``,0 where from (43)

JR(Kk,pm ,Sk−2`,p

m′

)=R→0

(iζ2)`J k,p ; k−2``,0 +O

((iζ2)m+1R−2`+2m+2 logm+2R

),

with m′ > m > l ; moreover

JR(Kk,pm ,Sk−2`,p

m′

)=

∫r=R

(Kk,pm ∂rS

k−2`,pm′ −Sk−2`,p

m′ ∂rKk,pm

)Rdθ.

Comparing (39) and (40), it is straightforward that the useful terms for computing J k,p ; k−2`,p arethe terms whose powers in r and in log r equal 0 in the product rKk,pm ∂rS

k−2`,pm′ and in the product

rSk−2`,pm′ ∂rK

k,pm .

From (22) and (25), straightforward calculi lead to

∂rSk,p(r, θ) =r k−1

∑j>0

(iζ2)j r2j

j∑n=0

logn r((k + 2j)Φk,pj,n (θ) + (n + 1)Φk,p

j,n+1(θ)), (45)

∂rKk,p(r, θ) =r−k−1

∑j>0

(iζ2)j r2j

j+1∑n=0

logn r((−k + 2j)Ψk,pj,n (θ) + (n + 1)Ψk,p

j,n+1(θ)), (46)

with the convention that Φk,pj,n = 0, ∀n > j and Ψk,p

j,n = 0, if (n > (j + 1)) or ((n > j) and k odd) or

((n > j) and k even and (j < k/2)). From (25) and (45), the terms of the product rKk,pm ∂rSk−2`,pm′ ,

whose powers in r and log r equal 0 are given by

(iζ2)`l∑j=0

∫ 2π

0

Ψk,pj,0 (θ)

[(k − 2j)Φk−2`,p

`−j,0 (θ) + Φk−2`,p`−j,1 (θ)

]dθ,

and similarly, from (22) and (46), the terms of the product rSk−2`,pm′ ∂rK

k,pm , whose powers in r and

log r equal 0 are

(iζ2)`l∑j=0

∫ 2π

0

Φk−2`,pl−j,0 (θ)

[(−k + 2j)Ψk,p

j,0 (θ) + Ψk,pj,1 (θ)

]dθ.

Making the difference of both contributions, we obtain (44).

RR n° 8204


4 Leading singularities and their first shadows in complex vari-ables

According to the previous section, the explicit calculation of both primal and dual singularities is essentialto obtain a precise description of the magnetic potential A for numerical purposes. The heuristics ofsuch a calculus have been described in subsections 3.1 and 3.2. It consists in deriving the shadow atany order of the leading singularities (primal or dual depending on which kind of singularities is beingconsidered). The difficulty of these calculations lies in the transmission conditions (continuity of thepotential A and of its normal derivative) across the boundary G, that have to be imposed on the angularfunctions Φ of the space Sλ. The use of appropriate complex variables like in [2] instead of the polarcoordinates (r, θ) simplifies drastically the calculations. Indeed, in our case, it will be sufficient to usean Ansatz using two complex variables z+ and z− (in S− and S+) involving only integer powers of z±,z±, log z±, and log z±. Thus, it is much less complex than a general Ansatz using polar coordinateswithout further information. The aim of this section is to derive the first shadow terms of both primaland dual leading singularities and to present a systematic method of calculus if more terms are needed.

4.1 Formalism in complex variables

Let us present the appropriate formalism in complex variables. Setting z = re iθ = x + iy it is obviousthat

∂z =1

2(∂x − i∂y ) and ∂z =

1

2(∂x + i∂y ),

from which we deduce

r∂r = z∂z + z∂z , ∂θ = i (z∂z − z∂z) and ∆ = 4∂z∂z . (47)

The leading terms sk,p0 = sk,p and kk,p0 = kk,p of the primal and dual singularities have a naturalexpression in complex variable: Setting for any nonnegative integer k :3

sk0(re iθ)

= sk,00 (r, θ) + i sk,10 (r, θ) and kk0(re iθ)

= kk,00 (r, θ)− i kk,10 (r, θ),

we obtain

∀k ∈ N sk0(z) = zk and

kk0(z) =

1

2kπz−k if k ∈ N \ 0,

k00(z) = −

1

2πlog z if k = 0.

(48)

The idea is to take advantage of formulas (47) and (48) to solve problems (19)–(20) and their analoguesfor dual functions, in order to determine shadow terms.

Complex variables in S− and S+. Intuitively, since the shadows sk,pj and kk,pj belong respectively toSk+2j and S−k+2j , the corresponding function of the complex variable skj and kkj should involve termsin logq z , q ∈ N. Note however that despite log(z) is well-defined in S− with its branch cut on R−, itis not defined in S+ (which contains R−).

To avoid such a problem we introduce two complex variables to perform the calculations with thesame determination of the complex logarithm. Since G is the broken line z : | arg(z)| = ω/2 wedefine z− and z+ as

z− = z, z+ = −z, (49)

3For uniformity of presentation we complete the sets of functions sk,p0 and kk,p0 by the convention that s0,10 = 0 and

k0,10 = θ.

Inria


which ensures that log(z−) and log(z+) are well defined respectively for z− ∈ S− and z+ ∈ S+, wherelog is the usual complex logarithm with its branch cut on R−. For the sake of simplicity, we simplydenote z− by z , and in (r, θ)–coordinates we have

z = re iθ, z+ = re iθ+ , where θ+ =

θ − π if θ ∈ (0, π],

θ + π if θ ∈ [−π, 0).(50)

As previously presented, the two primal leading singularities, sk,00 and sk,10 , are the respective real andimaginary parts of sk0(z) = zk , for k ∈ N, which writes sk0 = (χk0, ξ

k0 ) in the variables (z, z+) whereχ

k0(z) = zk , |arg z | 6

ω

2,

ξk0 (z+) = (−1)k(z+)k , |arg z+| 6 π −ω

2.

Shadows in complex variables For any k ∈ N and j > 1, we define the couples skj :=(χkj (z), ξkj (z+)

)and kkj :=

(ζkj (z), ηkj (z+)

)by

χkj (z) = sk,0j (r, θ) + i sk,1j (r, θ), and ζkj (z) = kk,0j (r, θ)− i kk,1j (r, θ), if |θ| 6ω

2,

ξkj (z+) = sk,0j (r, θ) + i sk,1j (r, θ), and ζkj (z+) = kk,0j (r, θ)− i kk,1j (r, θ), if |θ| >ω

2.

(51)

The interior problem (19) translates into ∂z∂zχkj = χkj−1 in S− and ∂z+∂z+ξkj = 0 in S+, and similarly

for (24). The transmission conditions (20) write equivalently in polar coordinates[r∂rs

k,pj

]G = 0 and[

∂θsk,pj

]G = 0 and similarly for kk,pj . As far as primal singularities are concerned, the degree of (quasi)

homogeneity of sk,pj is never 0 and the transmission conditions are equivalent to[∂zskj

]G = 0 and[

∂zskj]G = 0. The computation of the primal shadow terms of order j > 1 consists now in finding

(χkj (z), ξkj (z+)) such that

∂z∂zχkj = χkj−1 in S− , (52a)

∂z+∂z+ξkj = 0, in S+ , (52b)

∂zχkj + ∂z+ξ

kj = 0 on G , (52c)

∂zχkj + ∂z+ξ

kj = 0 on G . (52d)

If −k + 2j is not equal to 0, the dual shadow terms of order j > 1 are determined by solutions(ζkj (z), ηkj (z+)) to problem

∂z∂zζkj = ζkj−1 in S− , (53a)

∂z+∂z+ηkj = 0, in S+ , (53b)

∂zζkj + ∂z+η

kj = 0 on G , (53c)

∂zζkj + ∂z+η

kj = 0 on G . (53d)

Finally if −k + 2j vanishes, (ζkj (z), ηkj (z+)) satisfies

∂z∂zζkj = ζkj−1 in S− , (54a)

∂z+∂z+ηkj = 0, in S+ , (54b)

z∂zζkj − z∂zζkj − z+∂z+η

kj + z+∂z+η

kj = 0 on G , (54c)

ζkj − ηkj = 0 on G . (54d)

RR n° 8204


Performing the calculations, we will find that the following Ansatz spaces are the correct ones to solvethe above three problems: For λ ∈ Z and (`, n) ∈ N2 we define the R–vector spaces Z λ,±

`,n , Z λ`,n and

Sλ`,n by

Z λ,±`,n = SpanR

zλ−µ± zµ± logq(z±), with 0 6 µ 6 `, 0 6 q 6 n

, (55a)

Sλ,±`,n = Z λ,±`,n + Z λ,±

`,n , (55b)

Sλ`,n =u = (v(z−), w(z+)), with v ∈ Sλ,−`,n and w ∈ Sλ,+`,n

, (55c)

where Z λ,±`,n is the space of conjugates u with u ∈ Z λ,±

`,n . Note that Sλ`,n is a finite dimensional subspaceof the space Tλ introduced in (17).

Remark 4.1. We emphasize that the spaces Z λ,±`,n are R–vector spaces, and not C–vector spaces,

which will be important in the foreseen calculus.

We are going to prove the following theorem in the next subsections (for j = 1) and in Appendix A(for j > 2).

Theorem 4.2. Let k ∈ N and j > 1. Then the primal shadow skj in complex variables belongs to Sk+2jj,j .

If −k + 2j < 0, the dual shadow kkj belongs to S−k+2jj,j and if −k + 2j > 0, kkj belongs to S−k+2j

j,j+1 . The

functions sk,pj and skj (respectively kk,pj and kkj ) are linked bysk,0j (r, θ) = Re(

skj (re iθ)), kk,0j (r, θ) = Re

(kkj (re iθ)

), if k ∈ N

sk,1j (r, θ) = Im(

skj (re iθ)), kk,1j (r, θ) = − Im

(kkj (re iθ)

)if k ∈ N \ 0.

4.2 Calculation of the shadow function generated by zk and log z

In this section, we derive the shadow term generated by zk in the space Sk+21,1 for k ∈ Z \ −2 and in

S01,2 if k = −2. We will also find the shadow term generated by log z in S2

1,2. The shadows of thesefunctions are important since the leading term of the primal functions are the function zk with k ∈ N,while the leading term of the dual functions are log z and the functions z−k with k ∈ N \ 0.

Proposition 4.3. Let k ∈ Z \ −2, −1. A particular solution uk = (v k , w k) to the following problem

∂z∂zvk = zk in S− , (56a)

∂z+∂z+wk = 0, in S+ , (56b)

∂zvk + ∂z+w

k = 0 on G , (56c)

∂zvk + ∂z+w

k = 0 on G , (56d)

writes

v k(z) =sinω

π(k + 2)zk+2 log z +

sin(k + 1)ω

π(k + 1)(k + 2)zk+2 log z −

cosω

k + 2zk+2 +

zk+1z

k + 1(57a)

(−1)kw k(z+) =sinω

π(k + 2)zk+2

+ log z+ +sin(k + 1)ω

π(k + 1)(k + 2)zk+2

+ log z+ +cos(k + 1)ω

(k + 1)(k + 2)zk+2

+ . (57b)

Inria


Proof. Since the function zk+1z/(k + 1) satisfies

∂z∂z(zk+1z/(k + 1)

)= zk ,

we suppose that the solution (v , w) = (v k , w k) to (56) writesv(z) = azk+2 log z + a′zk+2 log z + bzk+2 + zk+1z/(k + 1),

(−1)kw(z+) = azk+2+ log z+ + a′zk+2

+ log z+ + b′zk+2+ .

(58)

Observe that equations (56a) and (56b) are satisfied by the Ansatz, for any complex numbers a,b, a′ and b′. Jump conditions will determine a, b, a′ and b′ in order (56) to be satisfied.

Equations (56c)–(56d) lead to four conditions for the coefficients a, b, a′ and b′, since bothequalities hold in arg(z) = ω/2 (i.e. arg(z+) = ω/2−π) and arg(z) = −ω/2 (i.e. arg(z+) = π−ω/2).For instance write equation (56c) in arg(z) = ω/2. Applying the following equalities

∀z ∈ S−, ∂zv(z) = a(

(k + 2)zk+1 log z + zk+1)

+ b(k + 2)zk+1 + zk z ,

∀z+ ∈ S+, ∂z+w(z+) = (−1)ka(

(k + 2)zk+1+ log z+ + zk+1

+

),

respectively in z = re iω/2 and z+ = re i(ω/2−π) and using constraint (56c) imply

∂zv |arg(z)=ω/2 + ∂z+w |arg(z+)=ω/2−π = r k+1e i(k+1)ω/2(

iπa(k + 2) + b(k + 2) + e−iω),

and similarly in z = re−iω/2 and z+ = re i(−ω/2+π):

∂zv |arg(z)=−ω/2 + ∂z+w |arg(z+)=−ω/2+π = r k+1e−i(k+1)ω/2(−iπa(k + 2) + b(k + 2) + e iω

).

Therefore ∂zv + ∂z+w vanishes on G iffiπa + b = −e−iω/(k + 2),

−iπa + b = −eiω/(k + 2),

hence

a =sinω

π(k + 2)and b = −

cosω

k + 2.

Very similar computations imply that the jump ∂zv + ∂z+w vanishes in θ = ω/2 iff

−(k + 2)e−i(k+1)ω/2iπa′ − (k + 2)e−i(k+1)ω/2b′ + ei(k+1)ω/2/(k + 1) = 0.

For θ = −ω/2, we have

(k + 2)ei(k+1)ω/2iπa′ − (k + 2)ei(k+1)ω/2b′ + e−i(k+1)ω/2/(k + 1) = 0.

Hence, ∂zv + ∂z+w vanishes on G iffiπa′ + b′ = ei(k+1)ω/(k + 1)(k + 2),

−iπa′ + b′ = e−i(k+1)ω/(k + 1)(k + 2),

from which we infer

a′ =sin(k + 1)ω

π(k + 1)(k + 2)and b′ =

cos(k + 1)ω

(k + 1)(k + 2).

Therefore, uk = (v k , w k) writes (57).

RR n° 8204


If k = −1, a particular solution to (56) necessarily involves the function z log z since ∂z∂z (z log z) =

z−1. We therefore have

Proposition 4.4. For k = −1, a particular solution u−1 = (v−1, w−1) to (56) writes

v−1(z) =sinω

πz log z +

ω − ππ

z log z − cosω z + z log z, (59a)

w−1(z+) = −sinω

πz+ log z+ −

ω

πz+ log z+ + z+. (59b)

Proof. Similarly to (58), for k = −1 we suppose that the solution (v , w) = (v k , w k) to (56) writesv(z) = az log z + a′z log z + bz + z log z,

w(z+) = −az+ log z+ − (a′ + 1)z+ log z+ − b′z+.

Very similar computations using the jump conditions (56c) on G imply necessarilyiaπ + b = −e−iω,

iaπ − b = e iω,

and using (56d), this leads to ia′π + b′ + 1 = i(ω − π),

ia′π − b′ − 1 = i(ω − π),

hence the proposition.

The case k = −2 is quite different since the source term belongs to T−2 of Lemma 3.1, hence thedegree of the shadow (as polynomial in log(r)) is bounded by 2 instead of 1. The following propositionshows that it is actually equal to 2.

Proposition 4.5. For k = −2, a particular solution u−2 = (v−2, w−2) to (54) writes

v−2(z) =1

2πsinω log2 z +

1

2πsinω log2 z −

1

π(sinω + (2π − ω) cosω) log z − z−1z (60a)

w−2(z+) =1

2πsinω log2 z+ +

1

2πsinω log2 z+ −

1

π(sinω + (π − ω) cosω) log z+

− cosω log z+ − cosω + (π − ω) sinω.(60b)

Proof. Suppose that the solution (v , w) = (v−2, w−2) to (56) for k = −2 writesv(z) = a log2 z + a′ log2 z + b log z − z−1z

w(z+) = a log2 z+ + a′ log2 z+ + b log z+ + b′ log z+ + c.

Then ∂zv = 2a z−1 log z + b z−1 + z−2z ,

∂z+w = 2a z−1+ log z+ + b z−1

+ ,

∂zv = 2a′ z−1 log z − z−1,

∂z+w = 2a′ z−1+ log z+ + b′ z−1

+ .

Inria


Writing the jump condition z∂zv − z∂zv − z+∂z+w + z+∂z+w = 0 on θ = ±ω/2, and since z+ = ze∓iπ

we infer the system 2iπ(a + a′) + (b − b + b′) = −2e−iω,

−2iπ(a + a′) + (b − b + b′) = −2e iω.

The continuity across G implies:

± 2iπ(a − a′) + (b − b − b′) = 0,

c − π2(a + a′) + πω(a + a′)∓ iπ(b − b′)∓ iω

2(b + b′ − b) + e∓iω = 0,

hencea = a′ =

1

2πsinω, b′ = − cosω,

and then

c = − cos(ω) + (π − ω) sinω, b = −1

πsinω +

ω − 2π

πcosω, b = −

1

πsinω +

ω − ππ

cosω.

In order to obtain explicit formulas of any first shadow terms, it remains to derive the shadow oflog z . The following proposition can be checked through straightforward calculations.

Proposition 4.6. A particular solution k01 = (ζ0

1 , η01) to (53) with the source term equal to−1/(2π) log z

(i.e. k = 0, j = 1) writes

−2πζ01 (z) =

sinω

4πz2 log2 z +

sinω

4πz2 log2 z

−sinω + 2π cosω

4πz2 log z −

3 sinω + 2(π − ω) cosω

4πz2 log z

−1

4(π sinω − cosω)z2 + zz log z − zz ,

(61a)

−2πη01(z+) =

sinω

4πz2

+ log2 z+ +sinω

4πz2

+ log2 z+

−sinω

4πz2

+ log z+ −3 sinω − 2ω cosω

4πz2

+ log z+

−1

4(3 cosω + (2ω − π) sinω) z2

+.

(61b)

4.3 Explicit expressions of the first shadows of primal singularities.

To obtain the expressions of the first shadows of the primal leading singularities, we just have to takethe real and imaginary parts of the functions uk given by Proposition 4.3, for any nonnegative integerk ∈ N:

sk,01 = Re(uk), sk,11 = Im(uk).

We recall that s−(r, θ) is defined for |θ| 6ω

2and s+(r, θ) for |θ| >

ω

2. We also recall from (50) that

θ+ = θ − π sgn θ.

RR n° 8204


The shadow sk,01 . For any k ∈ N, the even first order shadow of zk writes

sk,0−1 (r, θ) =(k + 1) sinω + sin(k + 1)ω

π(k + 1)(k + 2)r k+2

(log r cos(k + 2)θ − θ sin(k + 2)θ

)+ r k+2

(cos kθ

k + 1−

cosω cos(k + 2)θ

k + 2

),

(62a)

sk,0 +1 (r, θ) =

(k + 1) sinω + sin(k + 1)ω

π(k + 1)(k + 2)r k+2

(log r cos(k + 2)θ − θ+ sin(k + 2)θ

)+ r k+2 cos(k + 1)ω cos(k + 2)θ

(k + 1)(k + 2).

(62b)

The shadow sk,11 . For any k ∈ N \ 0, the odd first order shadow of zk writes

sk,1−1 (r, θ) =(k + 1) sinω − sin(k + 1)ω

π(k + 1)(k + 2)r k+2

(log r sin(k + 2)θ + θ cos(k + 2)θ

)+ r k+2

(sin kθ

k + 1−

cosω sin(k + 2)θ

k + 2

),

(63a)

sk,1 +1 (r, θ) =

(k + 1) sinω − sin(k + 1)ω

π(k + 1)(k + 2)r k+2

(log r sin(k + 2)θ + θ+ cos(k + 2)θ

)− r k+2 cos(k + 1)ω sin(k + 2)θ

(k + 1)(k + 2).

(63b)

4.4 Explicit expressions of the first shadows of dual singularities

The explicit expressions of the first shadows of the dual leading singularities are given for k ∈ N \ 0by

kk,01 =1

2kπRe(u−k), kk,11 = −

1

2kπIm(u−k),

where the functions uk are given by Propositions 4.3, 4.4 and 4.5 and for k = 0 we have

k0,01 = Re(k0

1),

where k01 is given by Proposition 4.6.

The dual shadow kk,01 , for k 6= 0.• For k > 3, the even first order shadow of the dual singularity z−k writes

kk,01 =1

2kπs−k,01 ,

Inria


where by extension, we denote by s−k,0 the formula (62), in which k is replaced by −k .• For k = 1, 2, we have

2πk1,0−1 (r, θ) =

sinω + ω − ππ

r (log r cos θ − θ sin θ)− cosω r cos θ + r (log r cos θ + θ sin θ) ,

2πk1,0 +1 (r, θ) =

sinω + ω

πr (log r cos θ − θ+ sin θ)− r cos θ,

4πk2,0−1 (r, θ) =

sinω

π

(log2 r − θ2

)−

1

π(sinω + (2π − ω) cosω) log r − cos 2θ,

4πk2,0 +1 (r, θ) =

sinω

π

(log2 r − θ2

+

)−

1

π(sinω + (2π − ω) cosω) log r − cosω + (π − ω) sinω.

The dual shadow kk,11 , for k 6= 0.• For k > 3, the odd first order shadow of z−k writes

kk,11 (r, θ) = −1

2kπs−k,11 ,

where we denote by s−k,1 the formula (63), in which k is replaced by −k .• For k = 1, 2, we have

−2πk1,1−1 (r, θ) =

sinω − ω + π

πr (log r sin θ + θ cos θ)− cosω r sin θ − r (log r sin θ − θ cos θ) ,

−2πk1,1 +1 (r, θ) =

sinω − ωπ

r (log r sin θ + θ+ cos θ) + r sin θ,

−4πk2,1−1 (r, θ) = −

1

π(sinω + (2π − ω) cosω) θ + sin 2θ,

−4πk2,1 +1 (r, θ) = −

1

π(sinω − ω cosω) θ+.

The shadow k0,01 .

−2πk0,0−1 (r, θ) =

sinω

2πr2(

cos 2θ(log2 r − θ2)− 2θ sin 2θ log r)

+(ω − 2π) cosω − 2 sinω

2πr2 (cos 2θ log r − θ sin 2θ)

+cosω − π sinω

4r2 cos 2θ + r2 log r − r2,

−2πk0,0 +1 (r, θ) =

sinω

2πr2(

cos 2θ(log2 r − θ2+)− 2θ+ sin 2θ log r

)+ω cosω − 2 sinω

2πr2 (cos 2θ log r − θ+ sin 2θ)

−3 cosω + (2ω − π) sinω

4r2 cos 2θ.

5 Numerical simulations

In order to illustrate the calculus of the first shadows, mostly what is proposed in Section 4, and thetechnique described in subsection 3.4 to compute the coefficients of expansion (21), we consider the

RR n° 8204


following problem (problem (1) with J = 0 except the non-homogeneous Dirichlet boundary condition)−∆A+ = 0 in Ω+,

−∆A− + 4iζ2A− = 0 in Ω−,

A =|θ|2π

on Γ,

[A]Σ = 0, on Σ,

[∂nA]Σ = 0, on Σ.(64)

Since the source term is even with respect to θ, the solutionA of (64) is θ-even. As a consequence, onlythe terms with indices p = 0 are involved in the Kondratev-type expansion (21). The computationaldomain Ω, depicted in Figure 2(a), is a disk of radius 50mm. We consider a conducting sector forω = π/4 (other values have been tested and the conclusions are similar). We particularly focus onthe behavior of the solution in the vicinity of the corner c. Parameter ζ is equal to 1/(5

√2)mm−1,

which corresponds to a physical skin depth of 5mm. The solution computed by the finite elementmethod, using P2 finite elements available in the library [7] and a mesh with 64192 triangles is plottedin Figure 2(b) for the real part and Figure 2(c) for the imaginary part.

ΓΩ+

Σ

Ω−c ω

(a) Domain Ω.

(b) Real part of the solution A. (c) Imaginary part of the solution A.

Figure 2: Domain Ω and the computed solution for problem (64).

First, we consider the computation of Λ0,0 by the use of formula (35) for the case m = 0 andm = 1. The value of A computed at the corner c is defined as the reference value for Λ0,0; note that

Inria


it is already a numerically approximated value. This reference value is compared to JR(K0,0m ,A) for

m = 0 and m = 1, a quantity which provides an approximate value of Λ0,0 by the method describedin subsection 3.4. Note that the convergence rate as a function of R is related to the first neglectedterms. The results are shown in Figure 3 and are consistent with the theoretical convergence rate(remind R0 defined in (26)). Note that concerning m = 1 and the smallest values of R, we canpresume that the discretization accuracy is attained: This should explain the behavior for the smallestvalues of R in Figure 3.

10−4 10−3 10−2

10−9

10−7

10−5

10−3

10−1

Radius R (m).

err0,0

m(R

).

m = 0

R20 log(R)m = 1

R40 log(R)

Figure 3: Accuracy for the computation of Λ0,0 as a function ofR.Quantity err0,0

m (R) is the relative error |JR(K0,0m ,A) −

A(c)|/|A(c)|.Reference value A(c): (0.114449904− i 0.0464907336).

Second, we also consider the computation of the coefficients Λ1,0 and Λ2,0. The reference valuesare taken from the computation of JR for the small value Rsmall = 5 · 10−5m of R. Results forJR(K1,0

m ,A), m = 0, 1 are shown in Figure 4; they are also consistent with the expected theoreticalbehaviors. Results for JR(K2,0

1 ,A) are shown in Figure 5. In order to deduce from JR(K2,01 ,A) the

approximate value of Λ2,0, we have to use the computed value for Λ0,0 and the coefficient J 2,0;0,0.This last coefficient can be obtained from Proposition 3.7:

J 2,0;0,0 = iζ2

∫ 2π

0

Ψ2,00,0(θ)

[4Φ0,0

1,0(θ) + Φ0,01,1(θ)

]−Ψ2,0

1,1(θ)Φ0,00,0(θ) dθ. (65)

Angular functions Φ0,00,0, Φ0,0

1,0 and Φ0,01,1 can be identified by comparing (18) and (22) when k = 0, p = 0,

and j ∈ 0, 1

s0,0j (r, θ) = r2j

j∑n=0

lognr Φ0,0j,n (θ),

RR n° 8204


and using s0,00 (r, θ) = 1 (5) and the expression of s0,0

1 (62)

Φ0,00,0(θ) = 1, Φ0,0

1,1(θ) =sinω

πcos 2θ, Φ0,0

1,0(θ) =

1−

cosω

2cos 2θ −

sinω

πθ sin 2θ for |θ| 6

ω

2,

cosω

2cos 2θ −

sinω

πθ+ sin 2θ for |θ| >

ω

2.

Angular functions Ψ2,00,0 and Ψ2,0

1,1 can be identified by comparing (23) and (25) when k = 2, p = 0 andusing (7) and k2,0

1 from Subsection 4.4

Ψ2,00,0(θ) =

1

4πcos 2θ, Ψ2,0

1,1(θ) = −1

4π2(sinω + (2π − ω) cosω).

Computing integrals, we obtain

J 2,0;0,0 = iζ2

(3√

2

4π+

5√

2

8

)≈ iζ21.221502.

10−4 10−3 10−2

10−6

10−5

10−4

10−3

10−2

10−1

100

Radius R (m).

err1,0

m(R

).

m = 0

R−1R20

m = 1

R−1R40

Figure 4: Accuracy for the computation of Λ1,0 as a function of R.Quantity err1,0

m (R) stands for the relative error |JR(K1,0m ,A) −

JRsmall(K1,0

1 ,A)|/|JRsmall(K1,0

1 ,A)|.Reference value JRsmall

(K1,01 ,A): (−12.970664− i 5.40915055).

Inria


10−4 10−3 10−2

10−5

10−4

10−3

10−2

10−1

100

Radius R (m).

err2,0

m(R

).

m = 1

Real partImaginary partR−2R4

0 log(R)

Figure 5: Accuracy for the computation of Λ2,0 as a function of R.Quantity err2,0

m (R) stands for the relative error of the real part, of the imaginarypart and of the modulus of (JR(K2,0

1 ,A)− JRsmall(K2,0

1 ,A)).Reference value JRsmall

(K2,01 ,A): (1406.54919 + i 4599.19999).

RR n° 8204


In Figure 6, we perform a qualitative description of the isovalues of A close to the corner comparingthe finite element solution and, successively,

• expansion (21) restricted to a composite order 1, i.e.

JRsmall(K0,0

1 ,A) + JRsmall(K1,0

1 ,A)s1,00 ,

• expansion (21) restricted to a composite order 2, i.e.

JRsmall(K0,0

1 ,A)(1+iζ2s0,01 )+JRsmall

(K1,01 ,A)s1,0

0 +(JRsmall

(K2,01 ,A)− J 2,0;0,0JRsmall

(K0,01 ,A)

)s2,0

0 ,

• expansion (21) restricted to a composite order 3, i.e. adding to the expression above the term(JRsmall

(K3,01 ,A)− J 3,0;1,0JRsmall

(K1,01 ,A)

)s3,0

0 ,

and replacing s1,00 by s1,0

0 + iζ2s1,01 . Computing integrals, we obtain

J 3,0;1,0 ≈ iζ21.522117 · 10−9 .

The reference value for JRsmall(K3,0

1 ,A) is (93037.6253− i 154720.669).

For the composite order 1, only the constant and linear terms with respect to r are collected in theKondratev-type expansion. For the composite order 2, the terms which behave as r2 and r2 log r areadded. Adding then the terms which behave as r3 and r3 log r leads to the composite order 3.

Both on the real and imaginary parts, we observe in Figure 6 as expected that the increase of theorder enables to increase the accuracy.

6 Conclusion

In this paper, we have provided corner asymptotics of the magnetic potential for the eddy currentproblem in a bidimensional domain. Such expansions involve two ingredients: The calculation ofboth primal and dual singularities, and the computation of the singular coefficients. Primal and dualsingularities of the non-homogeneous operator −∆ + iκµ0σ1Ω− in R2 are derived as infinite series,whose coefficients are obtained recursively. To compute the singular coefficients, we first tried toapply the method of moments, straightforwardly derived from the case of the Laplace operator. Thismethod is limited since it makes possible to obtain only the first singular coefficients of the cornerasymptotics and at a low order of accuracy, therefore we adapted the method of quasi-dual functionsintroduced in [3] to our specific problem. Numerical simulations of section 5 corroborate the theoreticalorder of accuracy obtained in the previous sections, illustrating the accuracy of the asymptotics.

Forthcoming work will deal with the high conducting case, (ı.e. the case σ goes to infinity),for which two small parameters appear: the distance to the corner and the skin depth

√2/(κµ0σ).

Preliminary results of the formal derivation of the magnetic potential have been obtained by Buret etal in [1], which have to be rigorously justified and extended.

Inria


0.244 X-0.0111 0.244 0.5

a_rmax_corner_r1 (0/1)

-0.0111 0.244 0.5

az (0/1) Y

Z

az (0/1)

X

Y

Z-0.0111 0.244 0.5


-0.0111 0.5

(a) Expansion restricted to composite order 1. Real part.

Y

-0.0301 0.00107

az (1/1)

Z

az (1/1)

-0.0613 -0.0301 0.00107


-0.0613 -0.0301 0.00107 X

Y

Z


-0.0613 -0.0301 0.00107 X-0.0613

(b) Expansion restricted to composite order 1. Imaginarypart.

-0.0111 0.5

az (0/1)

-0.0111 0.244 0.5


-0.0111 X

Y

Z0.244 0.5 X

Y

Z-0.0111 0.244 0.5

a_rmax_corner_r2 (0/1)az (0/1)

0.244

(c) Expansion restricted to composite order 2. Real part.

az (1/1)

X-0.0301

Y

Z0.00107

az (1/1)

-0.0613 -0.0301 0.00107


-0.0613 -0.0301 0.00107 X

Y

-0.0613 -0.0301 0.00107


Z-0.0613

(d) Expansion restricted to composite order 2. Imaginarypart.

0.244


Z

az (0/1)

-0.0111 X

Y

Z0.244 0.5


-0.0111-0.0111 0.244 0.5

az (0/1)

0.244 0.5 X

Y

-0.0111 0.5

(e) Expansion restricted to composite order 3. Real part.


0.00107


X

Y

Z-0.0613 -0.0301 0.00107 X

Y

Z-0.0613 -0.0301 0.00107

az (1/1)az (1/1)

-0.0613-0.0613 -0.0301 0.00107-0.0301

(f) Expansion restricted to composite order 3. Imaginarypart.

Figure 6: Comparison of the finite element solution and of the local expansion.

RR n° 8204


References

[1] François Buret, Monique Dauge, Patrick Dular, Laurent Krähenbühl, Victor Péron, Ronan Per-russel, Clair Poignard, and Damien Voyer. Eddy currents and corner singularities. IEEE Trans. onMag., 48(2):679–682, 2012.

[2] M. Costabel and M. Dauge. Construction of corner singularities for Agmon-Douglis-Nirenbergelliptic systems. Math. Nachr., 162:209–237, 1993.

[3] M. Costabel, M. Dauge, and Z. Yosibash. A quasidual function method for extracting edge stressintensity functions. SIAM Jour. Math. Anal., 35(5):1177–1202, 2004.

[4] M. Dauge, S. Nicaise, M. Bourlard, and J. Mbaro-Saman Lubuma. Coefficients des singularitéspour des problèmes aux limites elliptiques sur un domaine à points coniques. II. Quelques opérateursparticuliers. RAIRO Modél. Math. Anal. Numér., 24(3):343–367, 1990.

[5] Monique Dauge. Elliptic Boundary Value Problems in Corner Domains – Smoothness and Asymp-totics of Solutions. Lecture Notes in Mathematics, Vol. 1341. Springer-Verlag, Berlin, 1988.

[6] E.M. Deeley. Surface impedance near edges and corners in three-dimensional media. IEEE Trans.on Mag., 26(2):712–714, 1990.

[7] P. Dular and C. Geuzaine. Getdp: a general environment for the treatment of discrete problems.Source code. http://geuz.org/getdp/, 1997-2012.

[8] P. Grisvard. Boundary Value Problems in Non-Smooth Domains. Pitman, London, 1985.

[9] H. Haddar, P. Joly, and H.-M. Nguyen. Generalized impedance boundary conditions for scatteringproblems from strongly absorbing obstacles: the case of Maxwell’s equations. Math. ModelsMethods Appl. Sci., 18(10):1787–1827, 2008.

[10] V. A. Kondratev. Boundary value problems for elliptic equations in domains with conical or angularpoints. Trudy Moskov. Mat. Obšč., 16:209–292, 1967.

[11] V. A. Kondratev and O. A. Oleinik. Boundary-value problems for partial differential equations innon-smooth domains. Russian Math. Surveys, 38:1–86, 1983.

[12] V. A. Kozlov, V. G. Maz’ya, and J. Rossmann. Elliptic boundary value problems in domains withpoint singularities. Mathematical Surveys and Monographs, 52. American Mathematical Society,Providence, RI, 1997.

[13] M. A. Leontovich. Approximate boundary conditions for the electromagnetic field on the surfaceof a good conductor. In Investigations on radiowave propagation, volume 2, pages 5–12. PrintingHouse of the USSR Academy of Sciences, Moscow, 1948.

[14] V. G. Maz’ya and B. A. Plamenevskii. On the coefficients in the asymptotic of solutions of theelliptic boundary problem in domains with conical points. Amer. Math. Soc. Trans. (2), 123:57–88,1984.

[15] M.-A. Moussaoui. Sur l’approximation des solutions du problème de Dirichlet dans un ouvert aveccoins. In Singularities and constructive methods for their treatment (Oberwolfach, 1983), volume1121 of Lecture Notes in Math., pages 199–206. Springer, Berlin, 1985.

Inria

http://geuz.org/getdp/


[16] Serge Nicaise. Polygonal interface problems. Methoden und Verfahren der MathematischenPhysik, 39. Verlag Peter D. Lang, Frankfurt-am-Main, 1993.

[17] S. M. Rytov. Calcul du skin effect par la méthode des perturbations. Journal of Physics,11(3):233–242, 1940.

[18] T. B. A. Senior and J. L. Volakis. Approximate Boundary Conditions in Electromagnetics. IEEElectromagnetic Waves Series, 41. London : Institution of Electrical Engineers, Royaume-Uni,1995.

[19] S. Shannon, Z. Yosibash, M. Dauge, and M. Costabel. Extracting generalized edge flux intensityfunctions by the quasidual function method along circular 3-d edges. Preprint HAL available athttp://hal.archives-ouvertes.fr/hal-00725928., 2012.

[20] Barna A. Szabó and Zohar Yosibash. Numerical analysis of singularities in two dimensions. II.Computation of generalized flux/stress intensity factors. Internat. J. Numer. Methods Engrg.,39(3):409–434, 1996.

[21] S. Yuferev, L. Proekt, and N. Ida. Surface impedance boundary conditions near corner and edges:Rigorous consideration. IEEE Trans. on Mag., 37(5):3465–3468, 2001.

A Appendix: Tools for the derivation of the shadows at any order

As described previously, the singularities of the operator Lζ (13) are obtained as series involving theshadows of the leading singularities at any order. Moreover, these shadows can be described with thehelp of the complex variable Ansatz introduced in Section 4 and spaces Sλ`,n defined by equations (55).However, the derivation of a generic formula for the shadows is hardly (and tediously!) reachable,because of the growth of the number of the terms with the order of the shadow.

For instance, the leading singularity z ∈ S10,0 generates a primal shadow s1

1 ∈ S31,1. According to

Proposition 4.3, this shadow s11 contains 4 terms: z3 log z , z3 log z , z3 and z2z in the sector S−,

and we shall prove that each of these terms generates a shadow belonging to S52,2. This shows the

complexity of the generic expression of the singularities. Nevertheless, it is possible to obtain theexpansion of the singularities at any order using a formal calculus tool. The aim of this section is toprovide the elementary results in order to derive the singularities, after appropriate use of a formalcalculus algorithm.

Actually, we see that, generally speaking, the shadow of the k th leading singularity at the order jis a combination of functions of the form zk+2j−`z ` logn z (or its conjugate), where (n, `) ∈ N2 and(k, j) ∈ Z×N. Therefore, if we provide a way to obtain the shadow of this generic function, it will bepossible to assemble the terms in order to obtain the full expression of the singularities.

For any (λ, `, n) ∈ Z × N2, the idea of the derivation of the shadow generated by zλ−`z ` logn z

consists in performing the same calculus as for the first order shadows, by only considering the termsof higher degree in log z , the remainder terms being neglected, and treated then at the next iteration.The principle of such calculations holds behind the calculus of Section 4.

A congruence relation on the spaces Sλ`,n is thus needed. Moreover, due to the omission of the termsof lower degree, errors are generated on the boundary G, which have to be corrected and thereforeappropriate spaces of traces and a congruence on such spaces have also to be given.

RR n° 8204

http://hal.archives-ouvertes.fr/hal-00725928


A.1 Definition of the space Jλn on G and the congruence relation ≡ on Sλ`,n andJλn

Definition A.1. For (λ, n) ∈ Z× N denote by Jλn the space

Jλn =

g : ∀z ∈ G, g(z) = zλ

n∑q=0

logq z

αq, if arg z = ω/2

αq, if arg z = −ω/2, (αq)nq=0 ∈ Cn+1

. (66)

Remark A.2. We would obtain the same space Jλn if we replace everywhere zλ logq z in (66) by zλ logq z .The same holds with zλ logq z , or zλ logq z , instead of zλ logq z .

Definition A.3. The congruence relation on Sλ`,n (55) and Jλn (66) is defined as follows:

∀(λ, `, n) ∈ Z× N× N \ 0,

∀u, u′ ∈ Sλ`,n, u ≡ u′ iff u − u′ ∈ Sλ`,n−1 ,

∀g, g′ ∈ Jλn , g ≡ g′ iff g − g′ ∈ Jλn−1 ,

and for n = 0 we define ≡ as the usual equality between two functions.

Roughly speaking, the congruence relation consists in identifying two functions with the samecoefficient in front of the highest power of the logarithmic terms.

A.2 Generic elementary calculation

Throughout this subsection, we choose an integer λ ∈ Z (which corresponds to the degree of homo-geneity of the solution) and a natural number ` ∈ N. Define the sequences of functions of the complexvariables (f ∗n , g

∗n, h∗n)n∈N as

∀n ∈ N,

∀z ∈ S−, f ∗n (z) = zλ−2−`z ` logn z,

∀z ∈ G, g∗n(z) = zλ−1 logn z,

∀z ∈ G, h∗n(z) = zλ−1 logn z .

(67)

According to the previous observations, the primal and dual shadow terms at any order are generatedby linear combinations of such (f ∗n , g

∗n, h∗n). For any n ∈ N, we also denote by v ∗n the function

∀z ∈ S−, v ∗n (z) =

zλ−`−1

λ− `− 1

z `+1

`+ 1logn z, if ` 6= λ− 1,

1

n + 1

z `+1

`+ 1logn+1 z, if ` = λ− 1.

(68)

Now we choose n > 1 and investigate an induction step. Let (αn, βn, γn) ∈ R×C2, and let (fn, gn, hn)

be defined by∀z ∈ S−, fn(z) = αnf

∗n ,

∀z ∈ G, gn(z) =

βng

∗n(z), for θ = ω/2,

βng∗n(z), for θ = −ω/2,

∀z ∈ G, hn(z) =

γnh

∗n(z), for θ = ω/2,

γnh∗n(z), for θ = −ω/2.

(69)

Therefore,(fn, gn, hn) ∈ Sλ−2,−

`,n × Jλ−1n × Jλ−1

n .

Inria


We are going to show that the shadow term (Vn,Wn) generated by the triple (fn, gn, hn) can be obtainedrecursively in Sλ`+1,n+1 when λ 6= 0, and in S0

`+1,n+2 when λ = 0 by solving a partial differential equationproblem, thanks to the congruence relation ≡. Actually, the shadow (Vn,Wn) generated by (fn, gn, hn)

satisfies ∂z∂zVn = fn, in S− ,

∂z+∂z+Wn = 0, in S+ ,

∂zVn + ∂z+Wn = gn, on G ,∂zVn + ∂z+Wn = hn, on G .

(70)

Throughout this subsection, in order to simplify the notations, we set

`′ = λ− 2− `, m = λ− 1 . (71)

A.2.1 The case λ 6= 0

We first consider the case λ 6= 0, which corresponds to a source term which does not belong to T−2,with the notation of Lemma 3.1.

Proposition A.4. Let (λ, `, n) ∈ Z × N2 and let (fn, gn, hn) be defined by (69). We suppose thatλ 6= 0 and

either(` 6= λ− 1

), or

(` = λ− 1 and αn = 0

).

We define (vn, wn) asvn(z) = azλ logn+1 z + a′zλ logn+1 z + bzλ logn z ,

(−1)λwn(z+) = azλ+ logn+1 z+ + a′zλ+ logn+1 z+ + b′zλ+ logn z+ ,(72)

where, using notations (71)a =

1

πλ(n + 1)

(Imβn + αn

sinω(`+ 1)

(`+ 1)

)b =

1

λ

(Reβn − αn

cosω(`+ 1)

(`+ 1)

) ,

a′ =

1

πλ(n + 1)

(− Im γn + αn

sinω(`′ + 1)

(`′ + 1)

)b′ =

1

λ

(−Re γn + αn

cosω(`′ + 1)

(`′ + 1)

) .

Then, (αnv∗n + vn, wn) is a particular solution to (70) in Sλ`+1,n+1 for the congruence relation ≡. This

means that the couple of functions (Vn−1,Wn−1) defined by

Vn−1 = Vn − (αnv∗n + vn), Wn−1 = Wn − wn, (73)

satisfies the following problem:∂z∂zVn−1 = fn−1, in S− ,∂z+∂z+Wn−1 = 0, in S+ ,

∂zVn−1 + ∂z+Wn−1 = gn−1, on G ,∂zVn−1 + ∂z+Wn−1 = hn−1, on G ,

(74)

where(fn−1, gn−1, hn−1) ∈ Sλ−2,−

`,n−1 × Jλ−1n−1 × Jλ−1

n−1 .

RR n° 8204


More precisely we have the explicit formulas

fn−1 = −nαn`′ + 1

f ∗n−1, (75a)

gn−1 = n(b − a(n + 1)

((∓iπ)λ/2 + 1

)(∓iπ)

)g∗n−1 (75b)

− azmn−2∑q=0

((∓iπ)λ

(q

n + 1

)+ (n + 1)

(q

n

))(∓iπ)n−q logq z, for θ = ±ω/2,

hn−1 = −αne±iω(`′+1)

`′ + 1zm

n−1∑q=0

(q

n

)logq z (±iω)n−q (75c)

− b′zmn−1∑q=0

((±iπ)λ

(q

n

)+ n

(q

n − 1

))(±iπ)n−1−q logq z

− a′zmn−1∑q=0

((±iπ)λ

(q

n + 1

)+ (n + 1)

(q

n

))(±iπ)n−q logq z , for θ = ±ω/2.

For n = 0, the calculation is exact, meaning that f−1, g−1, and h−1 equal zero.

Before proving the above proposition, we first show that in the case:

` = λ− 1, and αn 6= 0,

we can reduce to Proposition A.4. A direct calculation based on formula (68) in the case when ` = λ−1

proves that there holds

Lemma A.5. If ` = λ− 1 and αn 6= 0, the shadow (Vn,Wn) solution to problem (70) writes

Vn = αnv∗n + Un,

where (Un,Wn) satisfies

∂z∂zUn = 0, in S− ,∂z+∂z+Wn = 0, in S+ ,

∂zUn + ∂z+Wn = gn − αnz−1 z`+1

`+ 1logn z, on G ,

∂zUn + ∂z+Wn = hn −αnn + 1

z ` logn+1 z, on G ,

(76)

As a consequence of Proposition A.4 and Lemma A.5 we find:

Corollary A.6. Let λ 6= 0, ` ∈ N and n ∈ N be chosen. Let (fn, gn, hn) be defined by (69). Then theshadow (Vn,Wn) generated by (fn, gn, hn) belongs to Sλ`+1,n+1. An analytic formula for (Vn,Wn) can berecursively deduced from Proposition A.4 and Lemma A.5.

Proof. If ` 6= λ− 1, we observe that problem (74) involves source terms whose power in log z is n− 1,therefore pushing forward the reasoning we can decrease the power of the logarithmic terms of thesource up to zero, and then solve exactly the last problem with no logarithm.

If ` = λ − 1, we first use Lemma A.5 and reduce to the situation of Proposition A.4 with thedifference that a real non-zero coefficient γn+1 then appears. Examining the formulas given in Propo-sition A.4, we can see that even with this non-zero γn+1, the solution (αnv

∗n + vn, wn) belongs to

Sλ`+1,n+1.

Inria


Proof of Proposition A.4. Observe that v ∗n satisfies

∂z∂zv∗n = f ∗n +

n

`′ + 1f ∗n−1,

∂zv∗n = z `

′ z `+1

`+ 1logn z + n

z `′

`′ + 1

z `+1

`+ 1logn−1 z and ∂zv

∗n =

z `′+1

`′ + 1z ` logn z.

To prove the proposition, we derive a particular solution (v , w) for the congruence relation ≡ to thefollowing problem

∂z∂zv ≡ 0, in S− ,∂z+∂z+w ≡ 0, in S+ ,

∂zv + ∂z+w ≡ −αnz `′ z `+1

`+ 1logn z + gn, on G ,

∂zv + ∂z+w ≡ −αnz `′+1

`′ + 1z ` logn z + hn, on G .

(77)

Note that if (vn, wn) given by (72) is a particular solution to (77) for the congruence relation ≡, thencareful calculations show that the couple (Vn−1,Wn−1) given by (73) satisfies (74), which will end theproof of the proposition.

Derive now a particular solution to (77) for the congruence relation ≡. For z = re±iω/2, thereholds

z `′ z `+1

`+ 1logn z =

e∓iω(`+1)

`+ 1zm logn z,

z `′+1 z `

`′ + 1logn z =

e±iω(`′+1)

`′ + 1zm logn z +

e±iω(`′+1)

`′ + 1zm

n−1∑q=0

(q

n

)logq z (±iω)n−q .

Hence, denoting by (c, c ′) the complex numbers given by

c = βn − αne−iω(`+1)

`+ 1, c ′ = γn − αn

e iω(`′+1)

`′ + 1,

the transmission conditions of (77) across G reads

∂zv + ∂z+w ≡ czm logn z for θ = ω/2 , (78a)

∂zv + ∂z+w ≡ czm logn z for θ = −ω/2 , (78b)

∂zv + ∂z+w ≡ c ′zm logn z for θ = ω/2 , (78c)

∂zv + ∂z+w ≡ c ′zm logn z for θ = −ω/2 . (78d)

Referring to Section 4.2, we assume that (v , w) writesv(z) = azλ logn+1 z + a′zλ logn+1 z + bzλ logn z ,

(−1)λw(z+) = azλ+ logn+1 z+ + a′zλ+ logn+1 z+ + b′zλ+ logn z+ ,(79)

with λ = m + 1. Then, the following properties hold on G:∂zv − aλzm logn+1 z − a(n + 1)zm logn z − bλzm logn z = bnzm logn−1 z ∈ Jmn−1 ,

(−1)λ∂z+w − aλzm+ logn+1 z+ − a(n + 1)zm+ logn z+ = 0 ∈ Jmn−1 ,

RR n° 8204


and similarly∂zv − a′λzm logn+1 z − a′(n + 1)zm logn z = 0 ∈ Jmn−1 ,

(−1)λ∂z+w − a′λzm+ logn+1 z+ − a′(n + 1)zm+ logn z+ − b′λzm+ logn z+ = b′nzm+ logn−1 z+ ∈ Jmn−1 .

Recall that z+ = −z . Due to the branch cut on R− of log, we have to distinguish the two branchesof G given by θ = ω/2 and θ = −ω/2. Actually, for θ = ±ω/2, since θ+ = ±ω/2 − ±π, we infer forn ∈ n + 1, n

zm logn z + (−1)λzm+ logn z+ = ±iπnzm logn−1 z − zmn−2∑q=0

(q

n

)(∓iπ)n−q logq z, for θ = ±ω/2.

Therefore

∂zv + ∂z+w = ±iπaλ(n + 1)zm logn z + bλzm logn z + gn−1, for θ = ±ω/2,

where gn−1 equals

gn−1 = bnzm logn−1 z − azmn−1∑q=0

((∓iπ)λ

(q

n + 1

)+ (n + 1)

(q

n

))(∓iπ)n−q logq z, for θ = ±ω/2.

We thus infer gn−1 ∈ Jmn−1. Therefore, the two first congruence relations in (78) are satisfied iff

iπaλ(n + 1) + bλ = c ,

−iπaλ(n + 1) + bλ = c .

This system of two equations has a unique solution (a, b) ∈ R2 given by

a =1

πλ(n + 1)Im(c) , b =

1

λRe(c) ,

hence

a =1

πλ(n + 1)

(Imβn + αn

sinω(`+ 1)

(`+ 1)

), b =

1

λ

(Reβn − αn

cosω(`+ 1)

(`+ 1)

).

Similar computations imply that

∂zv + ∂z+w = ∓iπa′λ(n + 1)zm logn z − b′λzm logn z + hn−1, for θ = ±ω/2,

where hn−1 equals

hn−1 = −αne±iω(`′+1)

`′ + 1zm

n−1∑q=0

(q

n

)logq z (±iω)n−q

− b′zmn−1∑q=0

((±iπ)λ

(q

n

)+ n

(q

n − 1

))(±iπ)n−1−q logq z

− a′zmn−1∑q=0

((±iπ)λ

(q

n + 1

)+ (n + 1)

(q

n

))(±iπ)n−q logq z , for θ = ±ω/2.

Inria


Therefore, the third and fourth equations in (78) write on G as

−iπa′λ(n + 1)zm logn z − b′λzm logn z ≡ c ′zm logn z , for θ = ω/2 ,

iπa′λ(n + 1)zm logn z − b′λzm logn z ≡ c ′zm logn z , for θ = −ω/2 .

Thus, the couple (a′, b′) is unique, in R2 and satisfies

iπa′λ(n + 1) + b′λ = −c ′ ,−iπa′λ(n + 1) + b′λ = −c ′ ,

hence

a′ =1

πλ(n + 1)

(− Im γn + αn

sinω(`′ + 1)

(`′ + 1)

), b′ =

1

λ

(−Re γn + αn

cosω(`′ + 1)

(`′ + 1)

),

and therefore, (vn, wn) given by (72) is a particular solution to (77), and satisfies the assertions of theproposition by constructions, which ends the proof.

A.2.2 The case λ = 0

We consider now the case λ = 0, which corresponds to a source term in T−2, according to Lemma 3.1.As seen previously at Proposition 4.5, it is necessary to increase the power of the logarithm terms ofthe solution by 1, in comparison with the case λ 6= 0. Moreover, observe that a constant term mayappear, as in Proposition 4.5 for instance, and it is necessary to consider problem (54) to ensure thecontinuity of the solution across G. The boundary data are now gn and hn+1 defined as

∀z ∈ G, gn(z) =

βn logn z, for θ = ω/2,

βn logn z, for θ = −ω/2,and hn+1(z) =

γn+1 logn+1 z, for θ = ω/2,

γn+1 logn+1 z, for θ = −ω/2,

where n ∈ N, βn ∈ C, and γn+1 ∈ C. The problem which we want to solve is, instead of (70),∂z∂zVn = fn, in S− ,∂z+∂z+Wn = 0, in S+ ,

z∂zVn − z∂zVn − z+∂z+Wn + z+∂z+Wn = gn on G ,Vn −Wn = hn+1 on G .

(80)

We have the following proposition.

Proposition A.7. Let (`, n) ∈ N × (N \ 0). Note that, according to (71), `′ = −2 − `. Define(vn, wn) as

vn(z) = a logn+2 z + a′ logn+2 z + b logn+1 z ,

wn(z+)= a logn+2 z+ + a′ logn+2 z+ + b′ logn+1 z+ ,(81)

where a =

1

2π(n + 2)

(1

n + 1

(Imβn + 2αn

sinω(`+ 1)

(`+ 1)

)+ Im γn+1

)b =

1

2

(1

n + 1

(Reβn − 2αn

cosω(`+ 1)

(`+ 1)

)+ Re γn+1

) ,

a′ =

1

2π(n + 2)

(1

n + 1

(Imβn + 2αn

sinω(`+ 1)

(`+ 1)

)− Im γn+1

)b′ =

1

2

(1

n + 1

(Reβn − 2αn

cosω(`+ 1)

(`+ 1)

)− Re γn+1

) .

RR n° 8204


We recall that according to (68), for λ = 0:

v ∗n = −1

(`+ 1)2(z/z)`+1 logn z,

Then, (αnv∗n + vn, wn) is a particular solution to (80) for the congruence relation ≡. This means that

the couple of functions (Vn−1,Wn−1) defined by

Vn−1 = Vn − (αnv∗n + vn), Wn−1 = Wn − wn, (82)

satisfies the following problem:∂z∂zVn−1 = fn−1, in S− ,∂z+∂z+Wn−1 = 0, in S+ ,

z∂zVn−1 − z∂zVn−1 − z+∂z+Wn−1 + z+∂z+Wn−1 = gn−1 on G ,Vn−1 −Wn−1 = hn on G ,

(83)

where(fn−1, gn−1, hn) ∈ S−2,−

`,n−1 × J0n−1 × J0

n.

More precisely we have the explicit formulas

fn−1 = −nαn`′ + 1

f ∗n−1, (84a)

gn−1 = (n + 2)a

n−1∑q=0

(q

n + 1

)(∓iπ)n+1−q logq z − (n + 2)a′

n−1∑q=0

(q

n + 1

)(±iπ)n+1−q logq z (84b)

− (n + 1)b′n−1∑q=0

(q

n + 1

)(±i(π − ω))n+1−q logq z −

nαne∓iω(`+1)

(`+ 1)2logn−1 z,

hn = a

n∑q=0

(q

n + 2

)(∓iπ)n+2−q logq z + a′

n∑q=0

(q

n + 2

)(±iπ)n+2−q logq z (84c)

+ b′n−1∑q=0

(q

n + 1

)(±i(π − ω))n+1−q logq z +

αne∓iω(`+1)

(`+ 1)2logn z.

Proof. Writing z∂zv`,n− z∂zv`,n and z+∂z+w`,n− z+∂z+w`,n and identifying the term in logn z , we infer

π(n + 2)(n + 1)(a + a′) = Imβn + 2αnsinω(`+ 1)

`+ 1,

(n + 1)(b + b′) = Reβn − 2αncosω(`+ 1)

`+ 1.

Then, writing the condition v`,n + αnv∗n − w`,n = hn+1 and identifying the term in logn+1 z , we obtain

a − a′ =1

π(n + 2)Im γn+1,

b − b′ = Re γn+1,

hence the proposition with gn−1 and hn given by (84b) and (84c).

Inria


For n = 0, we solve exactly the following problem

∂z∂zV`,0 =−α`,0

(`+ 1)2(z/z)`+1, (85a)

∂z∂zW`,0 = 0, (85b)

V`,0 −W`,0 =

γ1 log z + γ0, if θ = ω/2,

γ1 log z + γ0, if θ = −ω/2,(85c)

(z∂zV`,0 − z∂zV`,0)− (z+∂z+W`,0 − z+∂z+W`,0) =

β0, if θ = ω/2,

β0, if θ = ω/2,(85d)

thanks to the following proposition.

Proposition A.8. Let ` ∈ N. Recall that, in this case,

v ∗`,0 = −1

(`+ 1)2(z/z)`+1.

Define (v`,0, w`,0) asv`,0(z) = a log2 z + a′ log2 z + b log z ,

w`,0(z+)= a log2 z+ + a′ log2 z+ + b log z+ + b′ log z+ + c,(86)

where, settingd = β0 − 2α`,0e

−iω(`+1)/(`+ 1),

we have

a = a′ +1

2πIm γ1 =

1

4πIm(d + γ1),

b′ =1

2Re(d − γ1),

b =π − ωπ

b′ −α`,0

π(`+ 1)2sinω(`+ 1) + Im γ0,

b = b′ + b + Re γ1,

c = 2πa(π − ω)−α`,0

(`+ 1)2cosω(`+ 1)−

π − ω2

Im γ1 − Re γ0.

.

Then, (α`,0v∗`,0 + v`,0, w`,0) is a particular exact solution to (85).

Proof. The transmission conditions of (85) across G writes

(z∂zv`,0 − z∂zv`,0)− (z+∂z+w`,0 − z+∂z+w`,0) = ±2iπ(a + a′) + b − b + b′.

Since

d = β0 − 2α`,0e−iω(`+1)

`+ 1,

we infer that

2iπ(a + a′) + b − b + b′ = d ,

−2iπ(a + a′) + b − b + b′ = d .

RR n° 8204


Since

v`,0−w`,0 = a(±2iπ log z+π2) +a′(∓2iπ log z−2πω+π2) + (b− b−b′) log z± i(πb− (π−ω)b′)−c,

the continuity across G implies

± 2iπ(a − a′) + (b − b − b′) =

γ1, if θ = ω/2,

γ1, if θ = −ω/2,,

π2a + π(π − 2ω)a′ ± i(πb − (π − ω)b′)− c =α`,0

(`+ 1)2e∓iω(`+1) +

γ0, if θ = ω/2,

γ0, if θ = −ω/2,,

hence

a = a′ +1

2πIm γ1, b − b − b′ = Re γ1,

which ends the proof.

A.3 Formal derivation of the singularities

A.3.1 Principles

We are now ready to present how to obtain the singularities.

• First, choose the leading (primal or dual) singularity: zk , k ∈ Z or log z .

• For zk , use Proposition 4.3, Proposition 4.4 or Proposition 4.5 depending whether k ∈ −1,−2or not, and for log z use Proposition 4.6, in order to obtain the first shadow term. This shadowis composed of terms as zk+2−`z ` logn z , with n = 1 or 2 and ` ∈ 0, 1, 2.

• If k + 2 6= −2, check whether k + 2 − ` = 0, and apply recursively Proposition A.4 or Corol-lary A.5 for each term of the shadow. If k + 2 = −2, apply recursively Proposition A.7 and thenProposition A.8.

• Repeat the process till the desired order of the shadow.

• Sum all the terms.

A.3.2 Generic expression of the shadows at any order

The results of subsection A.2 leads to the following proposition, that provides the generic expressionof the primal singularities of Lζ.

Proposition A.9. Let k ∈ N. We recall that according to (48), sk0 = zk . For j > 1, the functionskj = (χkj , ξ

kj ) defined by (52), which is the shadow of order j of sk0, writes

χkj (z) = akjzk+2j logj z + a′kj z

k+2j logj z

+

j−1∑n=0

j−n∑m=0

bkj,nmzk+2j−mzm logn z +

j−1∑n=0

j−n∑m=0

ckj,nmzmzk+2j−m logn z , (88)

Inria


ξkj (z+) = (−1)kakjzk+2j+ logj z+ + (−1)ka′kj z

k+2j+ logj z+

+

j−1∑n=0

j−n∑m=0

b′kj,nmzk+2j−m+ zm+ logn z+ +

j−1∑n=0

j−n∑m=0

c ′kj,nmzm+ z

k+2j−m+ logn z+, (89)

where the coefficients akj , a′kj , bkj,nm, b′kj,nm, ckj,nm, and c

′kj,nm are real coefficients. These coefficients

are obtained by the induction process given by Proposition A.4.

From this proposition, explicit the primal singular functions Sk,p of corner asymptotics (4) can beeasily obtained. Actually, according to (18), it is sufficient to make explicit the functions sk,pj , whichare given by

sk,pj (r, θ) =

Re(χkj (re iθ)

)δp0 + Im

(χkj (re iθ)

)δp1 , if |θ| 6 ω/2,

Re(ξkj (re iθ+ )

)δp0 + Im

(ξkj (re iθ+ )

)δp1 , elsewhere ,

(90)

according to (51), where δp0 and δp1 are Kronecker symbols. We remind that θ+ is defined by

θ+ =

θ − π, if θ ∈ (0, π],

θ + π, if θ ∈ [−π, 0).

Then, we develop relation (18) with respect to the power of the logarithmic term log r , and we identifythe obtained expressions with the previous expression (22) of Sk,p(r, θ).

For instance, let us determine Φk,0j,n in the sector S−, for 0 6 n 6 j . Using (88), after tedious

calculus, we obtain:

Φk,0−j,j (θ) = (akj + a′kj) cos(k + 2j)θ , for n = j ,

and for any n 6 j − 1,

Φk,0−j,n (θ) =

(n

j

)θj−n(akj + a′kj) cos

((k + 2j)θ +

π

2(j − n)

)+

n∑q=0

(q

n − q

)θn−2q

j+q−n∑m=0

(bkj,n−q m + ckj,n−q m) cos(

(k + 2j − 2m)θ +π

2(n − 2q)

).

A similar reasoning leads to the generic expression of the dual singularities.

RR n° 8204

RESEARCH CENTREBORDEAUX – SUD-OUEST

351, Cours de la Libération

Bâtiment A 29

33405 Talence Cedex

PublisherInriaDomaine de Voluceau - RocquencourtBP 105 - 78153 Le Chesnay Cedexinria.fr

ISSN 0249-6399

Corner asymptotics of the magnetic potential in the eddy-current … · 2017. 1. 1. ·...

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